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Multiscale Lattice Boltzmann simulationsof droplet dynamics in time-dependent and

turbulent flows

Felix Milan

HPC-LEAPEUROPEAN JOINT DOCTORATES

This work is part of the European Union’s Framework Programme for Research and InnovationHorizon 2020 (2014 - 2020) under the Marie Sklodowska-Curie Grant Agreement No.642069.

Cover image: Sub-Kolmogorov droplet in a homogeneous and isotropic turbulent flow.

Thesis printed by Gildeprint Drukkerijen, Enschede.

A catalogue record is available from the Eindhoven University of Technology Library

ISBN: 978-90-386-4936-8

Copyright c© 2019 by Felix Milan, Eindhoven, The Netherlands.All rights are reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying,recording or otherwise, without prior permission of the author.

Multiscale Lattice Boltzmann simulationsof droplet dynamics in time-dependent

and turbulent flows

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. F.P.T. Baaijens, voor eencommissie aangewezen door het College voor

Promoties in het openbaar te verdedigenop woensdag 18 december 2019 om 13.30 uur

door

Felix Maria Niccolo Milan

geboren te Bamberg, Duitsland

Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de pro-motiecommissie is als volgt:

voorzitter: prof.dr.ir. G.M.W. Kroesen1e promotor: prof.dr. F. Toschi2e promotor: prof.dr. L. Biferale (Universita degli Studi di Roma “Tor Vergata”)copromotor(en): prof.dr. M. Sbragaglia (Universita degli Studi di Roma “Tor Vergata”)leden: prof.dr. P. Anderson

prof.dr. L. Brandt (KTH Stockholm)dr.ir. W.-P. Breugem (Technische Universiteit Delft)

Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven, is uitgevoerd inovereenstemming met de TU/e Gedragscode Wetenschapsbeoefening.

All details on simulations, data and scientific results shown in this thesis are recorded in thegitlab repository: https://git.phys.tue.nl/phd/milan/blob/master/README.md

https://git.phys.tue.nl/phd/milan/blob/master/README.md

Contents

1. Introduction 11.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Theoretical Background 52.1. The equations of fluid flows . . . . . . . . . . . . . . . . . . 5

2.2. Multicomponent flows . . . . . . . . . . . . . . . . . . . . . 6

2.3. Maffettone-Minale model (MM) . . . . . . . . . . . . . . . . 8

2.4. Turbulent flows . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1. The Richardson energy cascade . . . . . . . . . . . . 12

2.4.2. The Kolmogorov microscales . . . . . . . . . . . . . 14

3. The Lattice Boltzmann Method (LBM) 173.1. The Boltzmann and Lattice Boltzmann equation . . . . . . 17

3.2. The Chapman-Enskog Expansion (CEE) . . . . . . . . . . . 22

3.3. Shan-Chen multicomponent model . . . . . . . . . . . . . . 23

3.4. Validation tests for confined droplets . . . . . . . . . . . . . 24

3.5. A ghost node boundary flow scheme . . . . . . . . . . . . . 28

4. Droplet in a time-dependent flow: LBM and MM comparison 314.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2. Static droplet deformation . . . . . . . . . . . . . . . . . . . 33

4.3. Probing the parameter space: single component oscillatingshear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4. Multicomponent oscillating flow . . . . . . . . . . . . . . . . 39

4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5. Droplet breakup in a confined and time-dependent flow 515.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2. Simulation set up and definitions . . . . . . . . . . . . . . . 53

5.3. Droplet breakup in an oscillatory shear flow . . . . . . . . . 55

i

Contents

5.4. Inertial effects . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5. Confinement effects . . . . . . . . . . . . . . . . . . . . . . . 59

5.6. Flow topology . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6. LBM scheme for a sub-Kolmogorov droplet in HIT 676.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2. Structure of the algorithm . . . . . . . . . . . . . . . . . . . 68

6.3. Validation results . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7. Dynamics and breakup of a sub-Kolmogorov droplet in HIT 757.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2. Flow behaviour of the sub-Kolmogorov droplet . . . . . . . 76

7.3. Breakup predictions: LBM vs MM . . . . . . . . . . . . . . 77

7.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8. Conclusion and outlook 858.1. Droplets in laminar time-dependent flows . . . . . . . . . . 85

8.2. Sub-Kolmogorov droplets in turbulent flows . . . . . . . . . 86

8.3. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A. Numerical solution of the MM-model 89A.1. Simple shear flow . . . . . . . . . . . . . . . . . . . . . . . . 90

A.2. Uniaxial extensional flow . . . . . . . . . . . . . . . . . . . . 91

A.3. Planar hyperbolic flow . . . . . . . . . . . . . . . . . . . . . 92

B. A linearised MM-model solution for an oscillatory shear flow 95

Bibliography 97

Summary 111

Acknowledgement 113

Curriculum Vitae 115

Cover Illustration 117

ii

Contents

List of publications and conferences 119

iii

1. Introduction

1.1. Motivation

From natural occurring liquid aerosols, such as water droplets in cloudsand mist, to industrially produced emulsions, such as cremes and pastes,droplets can be found almost anywhere in our everyday lives. It is there-fore interesting to analyse the properties and in particular the dynamicsof droplet suspensions. The focus in this work will be on stable dropletsuspensions, i.e. the behaviour of two immiscible fluids, with the dispersedphase representing the droplet and the continuous phase the suspension.Droplet suspensions in liquids are called multicomponent systems, as theyare comprised of two fluid components, the droplet and its suspension. Onthe other hand, bubbles in water or droplet aerosols would be examples ofmultiphase systems. A particularly interesting problem involving droplets isthe mixing of two immiscible fluids, created by breakup of the droplets [1–4],which involves multiple scales and, in general, turbulent flow dynamics. Thelargest characteristic structures in turbulent flows are called eddies. In threedimensional turbulence the flow’s energy of the largest eddies is passed downto the smallest turbulent flow scales via a cascade mechanism, the Richard-son cascade [5–8]. The droplets are typically of the size of the smallest scales.Consequently, the droplets may still be subjected to large shear and/or stressrates due to the Richardson cascade, which transfers energy from the largesteddies to the smallest scales. The mixing of the two immiscible fluids isachieved via the breakup of the droplets into smaller ones. A very first studyon breakup of large droplets in turbulent flows was conducted by Hinze [9].Further studies on droplet deformation in turbulence followed, both exper-imentally and numerically [10–13]. Numerical studies investigating smallsub-Kolmogorov droplets, i.e. droplets with a characteristic length scalesmaller than the Kolmogorov scale, used more sophisticated approaches,such as the one by Cristini et al. [14] coupling a pseudo-spectral solver forthe outer flow with a boundary integral method to model the dynamics ofthe droplet. Furthermore Euler-Lagrangian approaches have been used to

1

1. Introduction

simulate sub-Kolmogorov bubbles in turbulent Taylor-Couette flows [15]. Aphenomenological model devised by Maffettone and Minale [16] has beenused very effectively in conjunction with pseudo-spectral flow solvers tomodel the statistics of sub-Kolmogorov droplet dynamics and their breakupconditions [17, 18]. The Maffettone-Minale model is an extension to Tay-lor’s theory, a perturbative approach to a droplet in Stokes flow [19–21].Even in the absence of turbulence, droplet dynamics are complex flows toinvestigate and model [22–25], especially in the presence of strong confine-ment [26–38]. Theoretical models accounting for the effect of confinementexist as well, such as the Shapira-Haber model [39, 40] extending Taylor’stheory [19, 20] and an extension to the Maffettone-Minale model [16] in-cluding the effect of confinement [41,42]. Both the Shapira-Haber and theconfined Maffettone-Minale model were experimentally validated [43, 44].Further factors affecting the flow of multicomponent systems are the roleof inertia [12,25,45–64], varying startup conditions [65–69] and the dropletdynamics in laminar time-dependent flows [70–74]. In order to model ourmulticomponent droplet system we use Lattice Boltzmann methods (LBM),which are a useful computational tool for the description of general mul-tiscale flow problems. LBM’s fame is due to its ability to couple physicalphenomena on the microscale, such as interface dynamics, polymers andthermal fluctuations, and continous hydrodynamical flows at the macro-scopic scale, e.g. flow-structure coupling and capillary fluctuations. LBMare derived from physical laws on the mesoscale, but can describe macro-scopic flow phenomena via a coarse grained procedure starting from proba-bility distribution functions at the kinetic level [75, 76]. This makes LBMvery successful in simulating the physics of fluids over a very broad rangeof scales. A further advantage of using LBM for multicomponent systemsis a diffuse interface model called the Shan-Chen multicomponent model(SCMC) [77,78], which is able to simulate the two-way coupling between thefluids, the droplet and the solvent. The nature of the SCMC model in theLBM framework made it possible to accommodate complex dynamics on themicroscopic level, such as non-ideal effects [79], coupling with polymer micro-mechanics [80] and thermal fluctuations [81,82]. This LBM multicomponentscheme has also been used widely for the modelling of breakup behaviour inlaminar flows [12,55,83–90]. In this thesis we will study the effect of generictime-dependency on droplet dynamics and breakup in laminar flows andthen move on to the study of sub-Kolmogorov droplet dynamics in homoge-

2

1.2. Thesis outline

neous and isotropic turbulence (HIT). We find that breakup in confined andoscillatory shear flows is not only dependent on the degree of confinement,but also on the flow start up conditions and the explicit time-dependencyof the solvent flow. For all problems treated in this work we use a newlydeveloped LBM boundary flow scheme, and compare the LBM simulationresults with the Maffettone-Minale model [16] and with other theoreticalpredictions, whenever suitable.

1.2. Thesis outline

The thesis is organized according to the following chapters. Chapter 1 con-cerns the general motivation of the studies of droplets in turbulent andcomplex flows. Chapter 2 gives a brief introduction to droplet flow modelsstarting from perturbative models in the Stokes limit to phenomenologicalmodels, which are valid for generic flow fields. Furthermore, a brief intro-duction to turbulent flows and the Kolmogorov scales is given. Chapter 3provides an overview of LBM and in particular SCMC, an extended LBMdiffuse interface model. Chapter 4 discusses droplet behaviour in a low am-plitude time-dependent shear flow, which is extended in chapter 5 to largeamplitude droplet dynamics and breakup in time-dependent flows with andwithout confinement. Chapter 6 outlines the details of a multiscale LBMalgorithm to model sub-Kolmogorov droplet dynamics and chapter 7 com-pares the breakup predictions of the Maffettone-Minale model [16] with theLBM predictions. In the end, chapter 8 provides an overall conclusion to thethesis and an outlook to further research on the sub-Kolmogorov dropletdynamics in homogeneous isotropic turbulence.

3

2. Theoretical Background

This chapter provides an introduction to the theoretical background nec-essary for the proceeding chapters. Firstly, we discuss the general Navier-Stokes and Stokes equations and then introduce models for multicomponentsystems in Stokes flow. We also discuss in detail a phenomenological dropletdeformation model, the Maffettone Minale [16], due to its adaptability todroplet dynamics in arbitrary flows. In the end, we give a brief overviewof the phenomenology of turbulent flows with an emphasis on the energycascade and the dissipative scales.

2.1. The equations of fluid flows

The governing equation for incompressible fluid flows are the Navier-Stokesequations (NSE):

∇ · v = 0

∂tv + (v · ∇)v = −1

ρ∇p+ ν∇2v (2.1)

The first equation represents the mass conservation of the incompressiblefluid, while the second describes the change of the fluid’s momentum dueto pressure and dissipative forces. v(x, t) represents the velocity field of thefluid at location x and time t, p is the pressure of the flow, ρ the densityof the fluid and ν represents the kinematic viscosity. We can define theReynolds number Re in order to provide an order of magnitude estimate ofthe ratio of the inertial (v · ∇v) and viscous term (ν∇2v) in equation (2.1)

Re ≡ LcUcν

(2.2)

with Lc and Uc being a characteristic length and velocity scale of the flowrespectively [6]. Using the characteristic length and velocity scales Lc and

5


Uc in addition to the kinematic viscosity ν we can non-dimensionalise theNSE:

∂tv + (v · ∇)v = −∇p+1

Re∇2v (2.3)

where˜denotes that the physical quantity is given in units of Lc, Uc andν. In the limit of low Reynolds numbers, i.e. Re→ 0, equation (2.3) reducesto the Stokes equation

−∇p+ η∇2v = 0 (2.4)

where we use the dimensional quantities p and v again. η = ρν is thedynamic viscosity of the flow. Both the non-linear term ∼ v · ∇v and thetime dependency ∂tv of equation (2.3) vanish. Thus, the Stokes equationis a stationary linear differential equation, whose solutions are unique [91].This is of great importance in order to find analytical solutions to weaklydeformed droplets in Stokes flow, as will be explained in the next section.

2.2. Multicomponent flows

We have derived the Stokes equation (2.4) in section 2.1 as a limiting case ofthe Navier-Stokes equations (2.1) for Re→ 0. We reformulate equation (2.4)for a two component system, where the subscript d denotes the droplet andthe subscript s the solvent component:

−∇ps + ηs∇2vs = 0

−∇pd + ηd∇2vd = 0 (2.5)

In order to solve (2.5) we employ a perturbation scheme developed byTaylor [19, 20], where we expand the pressure p and the velocity v up tolinear order in the capillary number Ca for both the droplet d and externalflow phase s (subscripts are omitted).

p = peq + Ca pT +O(Ca2)

v = veq + CavT +O(Ca2) (2.6)

6

2.2. Multicomponent flows

with Ca being the capillary number, which serves as a control parameterfor the droplet deformation. peq and veq are the pressure and velocity ofthe undeformed droplet respectively. Ca is given by the ratio of viscous andinterfacial forces of the droplet

Ca =ηsRG

σ(2.7)

where G is the shear rate of the solvent flow vs, defined via G ≡ ‖∇vs‖,R is the radius of the undeformed droplet, ηs the dynamic viscosity of thesolvent and σ the surface tension between the two liquids at their interface.The interpretation of the capillary number can be deduced from the Stokesequations (2.5) of the two component system: it represents the ratio of theviscous shear stresses in the flow ∼ ηs to the surface tension σ. The extentof deformation of the spherical liquid droplet can then be given by

D =L−WL+W

(2.8)

where L and W are the major and minor axis of the deformed ellipsoidaldroplet respectively. With the help of Taylor’s theory we may now determinethe deformation parameterDT of the droplet up to first order in the capillarynumber Ca [19,20]

DT = Ca16 + 19χ

16 + 16χ(2.9)

with

χ =ηdηs

(2.10)

being the viscosity ratio of the two immiscible fluids, where ηd is thedynamic viscosity of the droplet. Since the Taylor deformation theory con-siders the droplet to be unbounded, we employ an extension to Taylor’stheory [19, 20], the Shapira-Haber model (SH-model) [39, 40], which hasa similar line of reasoning as Taylor’s theory, but takes the issue of con-finement into account. Therefore, in the Shapira-Haber model [39, 40] thedeformation parameter is given by

DSH = Ca16 + 19χ

16 + 16χ

[1 +

(R

Lc

)3]Cs

52χ+ 1

χ+ 1(2.11)

7


where Cs is the so called shape factor [40].

2.3. Maffettone-Minale model (MM)

A more refined model than the one proposed by Taylor [20, 21] is a phe-nomenological model developed by Maffettone and Minale [16] (to be refer-enced as MM-model from here on). In the MM-model the droplet is assumedto be always ellipsoidal, so that we can describe it via a second rank tensorMij

Mij =1

VD

∫VD

d3x

(δij −

xixj‖x‖2

)(2.12)

which is also referred to as the morphology tensor. VD denotes the volumeof the droplet and x the distance from the droplet’s centre of mass to apoint inside VD. Droplet deformations are characterised via the componentsof Mij , for example an undeformed droplet has a morphology tensor ofMij = δij . It should be noted, that the MM-model is able to model dropletdynamics in generic flows and is not limited to the special case of Stokesflow. The time evolution of Mij due to an external flow field is given by theMM equation:

dMij

dt= Ca [f2(SikMkj +MikSkj) + ΩikMkj −MikΩkj ]−f1

(Mij − 3

IIIMIIM

δij

)(2.13)

where Sij is the strain-rate and Ωij the rotation-rate tensor of the solventflow, which are the symmetric and anti-symmetric parts of the velocitygradient tensor ∂jvi respectively:

Sij =1

2(∂jvi + ∂ivj) (2.14)

Ωij =1

2(∂jvi − ∂ivj) (2.15)

IIIM = det(Mij) and IIM ≡ 12(M2

kk −MijMij) are the third and secondtensor invariants of Mij . Unlike previous analytical approaches to modeldroplet deformation in laminar flows [20] the MM-model is not based on aperturbative expansion in the capillary number Ca. Thus we can increase

8

2.3. Maffettone-Minale model (MM)

Ca to relatively large values in our LBM simulations, since we have a robustanalytical model to compare it with. However, it should be noted, that theMM-model requires the droplet shape to be ellipsoidal at all times (an adhoc assumption). Since we want to compare the LBM simulations with theMM-model, we need to make sure that we remain in the linear flow regimeand check that our deformed LBM droplet is actually ellipsoidal at all times.We remark that the MM-model has to be modified to account for a confineddroplet, as was the case for Taylor’s theory, see equation (2.11). This can beachieved by modifying the parameters f1 and f2 in equation (2.13) for theconfined case. In the unbounded case [16], which we call MM-unbounded,we have

fun1 (χ) =

40(χ+ 1)

(3 + 2χ)(16 + 19χ)

fun2 (χ,Ca) =

5

3 + 2χ+

3 Ca2

2 + 6 Ca2 (2.16)

and for the confined case [41] which we call MM-confined

f1(χ, α) =fun

1 (χ)

1 + f c1(χ)Csα3

8

f2(χ,Ca, α) = fun2 (χ,Ca)

(1 + f c2(χ)Cs

α3

8

)(2.17)

with

f c1(χ) =44 + 64χ− 13χ2

2(1 + χ)(12 + χ)

f c2(χ) =9χ− 10

12 + χ(2.18)

Cs denotes a form factor, which depends on the degree of confinement [39,40], and

α =2R

Lz(2.19)

9


is the aspect ratio of the droplet length scale, the droplet diameter 2R, tothe scale of the confinement Lz, e.g. the width of a channel, see figure 3.3.The MM-model is originally designed for highly viscous flows, i.e. flows whichare close to the Stokes limit (see equation 2.4). Nevertheless, we will see inchapter 7 that the MM-model can be used to study droplet deformation inturbulent flows on the sub-Kolmogorov scale, as the limit of highly viscousflows is still applicable on this scale, see section 2.4.

2.4. Turbulent flows

So far we have only dealt with laminar flows, be it for single or multicompo-nent systems. However, most flows in nature are not laminar but turbulent.Turbulent flows show specific characteristics which are [5]:

1. Irregularity: Turbulent flows are inherently unpredictable, which ismost evident by the absence of a general solution to the Navier-Stokesequations in the turbulent regime. Nevertheless, one might think thatthere might be an effective theory of turbulence even in the absenceof a general solution to the NSE. However, this is not the case, asfor example the distance s(t) between the trajectories of two tracerparticles in a turbulent flow shows choatic behaviour, because it ishighly sensitive to its initial condition. This can be expressed by theLyapunov exponent [92]:

s(t) = s0eλLt (2.20)

with s0 = s(t = 0). λL > 0 is the Lyapunov exponent for the dis-tance s(t) between the two tracer particle trajectories. We see fromequation (2.20), that a slight difference in the initial conditions s0,leads to an exponential growth of the distance s(t) between the twotracer particle trajectories according to equation (2.20), even if s0 isinfinitesimally small. Due to this sensitivity on initial conditions [92],statistical models are used to investigate turbulent flow properties [8].

2. Diffusivity of energy: The transfer of energy from large scales tosmaller ones, the so called Richardson cascade, can be seen as themost iconic characteristic of turbulence. The transfer of energy is also

10


responsible for rapid mixing in case of a multicomponent turbulentflow.

3. Large Renynolds numbers Re: For large Re the non-linearities ofthe NSE are dominant, which causes the irregularity of the flow andthus the absence of an analytical solution to the NSE. In this regimeonly statistical models can adequately describe the flow.

4. Three dimensional flow: The transfer of energy from larger tosmaller scales, the Richardson cascade, is only possible due to thevortex stretching and elongation in three dimensions 1.

5. Dissipative flow: At the smallest scale of the Richardson cascade, theKolmogorov scale, the viscous shear stresses dominate and dissipateenergy. Every turbulent flow requires an energy input at the largestscales to compensate for the dissipated energy at the Kolmogorovscale.

In general, it is useful to decompose the velocity field v(x, t) into a meanflow and its fluctuations [5]:

v(x, t) = vm(x) + v(x, t) (2.21)

where the mean flow field vm(x) is the time average of the velocity field

vm(x) = 〈v(x, t)〉t (2.22)

with 〈. . .〉t denoting a time average. In this thesis we only deal withhomogeneous and isotropic turbulence (HIT), for which the mean flow isvm(x) = 0 and we are only left with the velocity fluctuations v(x, t), whichwe shall also call the velocity field from here on. Homogeneous states thatthe flow is “statistically” invariant under spatial translations of

v(x, t) 7→ v(x + a, t) (2.23)

with a ∈ R3 being an arbitrary change in position [6]. Analogously,isotropic means that the flow is “statistically” invariant under arbitraryrotations of1It should be noted, that a cascade exists in 2D turbulence as well. However energy is

transferred from the smallest to largest scales, so that the 2D turbulence cascade isoften called an inverse cascade [6, 8].

11


v(x, t) 7→ A · v(x, t) (2.24)

with A ∈ SO(R3), where SO(R3) denotes the special orthogonal transfor-mation group in R3 [6]. Therefore, the HIT turbulent flow possesses bothtranslational and rotational symmetry. These symmetries are of great im-portance in order to discuss the phenomenology of turbulence. There aretwo main phenomena of turbulence, which are central to this work: for onethe Richardson cascade or turbulence cascade is the underlying mechanismof energy transfer in a turbulent flow and ultimately responsible for differ-ent flow behaviour on different length scales. Secondly, we will discuss theimportance of the dissipative scales, the Kolmogorov microscales and evensmaller scales, the sub-Kolmogorov scales.

2.4.1. The Richardson energy cascade

We have mentioned briefly that turbulent flows are dissipative. Yet, inmany examples of turbulence structures appear to be stable [5, 6, 8]. Howcan this be the case for a dissipative flow? In 1922 Louis Fry Richardsongave an explanation to this apparent paradox [6], which is the Richardsonor energy cascade: Every fully developed turbulent flow passes its energyfrom its largest scale of size l0, to its smallest scale of size ηK via a cascademechanism, see figure 2.1. l0 is taken to be the size of the largest eddies,with an eddy being loosely defined as the largest turbulent structure in theflow. We see from figure 2.1 that the energy is transferred via the energydissipation ε to different intermediate length scales of sizes rnl0, wheren ∈ N and 0 < r < 1, before reaching the dissipative scale ηK . The regionwith scales of length ηK < ln < l0, with ln = rnl0, is called the inertialsubrange. Two key assumptions of the Kolmogorov theory of turbulence(1941) are fulfilled by the cascade mechanism [6]: The inertial subrange isstatistically scale invariant, i.e. physical properties are merely rescaled bymoving from larger to smaller scales. Secondly, the cascade gives rise tolocality of interactions, e.g. if we deal with a length scale ln = rnl0, it hasreceived energy from the larger scale ln−1 = rn−1l0 and its energy is passedon to a smaller scale of size ln+1 = rn+1l0.

12


K

Figure 2.1.: Sketch of the Richardson energy cascade: the energy of the turbulent floware transferred through a cascade mechanism from the size l0 of the largest eddies, theintegral scale, to the smallest scale ηK , where energy starts to dissipate. The constraintson the intermediate lengths of size rnl0, with n ∈ N, are 0 < r < 1. The blobs on eachlength scale level represent the respective eddies of that scale, with l0 being the length ofthe largest eddies in the flow. Courtesey of [6].

13


2.4.2. The Kolmogorov microscales

We have seen that the energy cascade transfers energy from the largesteddies of size l0 to smaller scales. We now investigate the dissipative scales,the Kolmogrov microscales, at which energy will not only be transferred butstarts to dissipate due to the kinematic viscosity ν. In order to investigate theimportance of energy dissipation on the microscales, we Fourier transformthe NSE (2.1) and investigate its spectral properties. The velocity field canbe expressed in spectral space via a set of mode vectors k.

v(x, t) =∑k

v(k, t) e−ik·x (2.25)

and the Fourier transformed velocity field in spectral space is given by

v(k, t) =1

L3x

∫ALx

d3x v(x, t) eik·x (2.26)

with ALx = Lx × Lx × Lx, where Lx ∈ R is the length scale of thefinite periodic real space ALx for the velocity field v(x, t) [6]. The Fouriertransformed incompressible Navier-Stokes equations (FTNSE) [7] in spectralspace are then given by

∂tvj(k, t)+νk2vj(k, t) = −ikm

(δjl −

kjklk2

)∑k′

vl(k′, t)vm(k−k′, t) (2.27)

with δjl being the Kronecker Delta and k2 = kjkj . The incompressibilitycondition in Fourier space kjvj(k, t) = 0 has been implicitly used. Aftercontracting equation (2.27) with the velocity field vj(k, t), we obtain anenergy balance equation [7]

∂E(k, t)

∂t= T (k, t)− 2 νk2E(k, t) (2.28)

with E(k, t) ≡ 12vj(k, t)vj(k, t) being the kinetic energy of the flow and

T (k, t) an energy transport term. Equation (2.28) represents the transportof energy of the larger scales to smaller scales according to the procedure ofthe Richardson cascade, until energy dissipates on the Kolmogorov scale andsub-Kolmogorov scale, where the kinematic viscosity ν dominates. Thus, forscales in the inertial subrange, i.e. k 1

ηK, only the transport of energy is

14


relevant and the dissipative term in equation (2.28) can be neglected. If wesum over all k modes, thus averaging over all length scales, equation (2.28)reads ⟨

dE

dt

⟩t

= −ε (2.29)

where

E ≡∑k

E(k, t) =1

2

∑k

vj(k, t)vj(k, t) (2.30)

is the mean energy of the flow per unit mass and

ε ≡ 2ν

⟨∑k

k2E(k, t)

⟩t

(2.31)

is the mean energy dissipation of the flow per unit mass [6]. The transport

function vanishes∑k

T (k, t) = 0, because the energy is only transported via

the energy cascade. We can now define the dissipative scale in turbulentflows, the Kolmogorov scale: The energy input is given by the energy dissi-pation ε according to equation (2.29). Since we are now in the dissipativeregion in terms of the Richardson cascade, the transferred energy from largerscales dissipates according to the kinematic viscosity ν. Consequently, wecan define a length ηK , time tη and velocity scale uη through dimensionalanalysis

ηK =

(ν3

ε

) 14

tη =

√ν

ε

uη = (νε)14 (2.32)

which are the Kolmogorov microscales. An alternative definition of thedissipative scale, is the local Reynolds number of the Kolmogorov scale

Reη ≡ηKuην

= 1 (2.33)

15


At this point we should also consider the Taylor Reynolds number [6]

Reλ =E

ν

√10

3ΩE(2.34)

with

ΩE ≡1

2L3x

∫ALx

d3x ω(x, t) · ω(x, t) (2.35)

being the mean enstrophy per unit mass, where

ω(x, t) = ∇× v(x, t) (2.36)

is the vorticity of the flow. Enstrophy can be seen as a measure of vorticesand rotation in a turbulent flow [8]. Since Reλ is only defined via globalflow quantities and not specific length or velocity scales, it is an appropriatemeasure for the degree of turbulence of a flow. Furthermore, it has to bestressed that the Kolmogorov microscale phenomenology still holds true forlength scales l < ηK , where ε acts as an energy input from larger scales andν is responsible for its dissipation. In this work we deal with sub-Kolmogorovlength scales for droplet dynamics, whose radii R fulfil R ηK , for whichthe local Reynolds number Re < 1.

16

3. The Lattice Boltzmann Method(LBM)

The equations for incompressible fluid flows, the Navier-Stokes equations (2.1),are continuum equations, i.e. they contain continuous functions, the scalardensity ρ(x, t) and pressure p(x, t) fields and the fluid velocity field v(x, t).In this chapter we will show that a discrete microscopic description, theLattice Boltzmann equation (LBE), can be used to derive the equationsof fluid flows (equation (2.1)) in the hydrodynamic limit. We will also out-line the general algorithmic structure of the Lattice Boltzmann methods(LBM). Furthermore, we will introduce the Shan-Chen multicomponentmodel (SCMC) [77, 78], which is an LBM based diffuse interface model.Moreover, we report validation results for the SCMC model and introducean LBM ghost node boundary scheme to be used in all simulations per-formed in chapters 4, 5, 6 and 7.

3.1. The Boltzmann and Lattice Boltzmann equation

The Boltzmann equation and Lattice Boltzmann equation can be derivedfrom Hamiltonian classical mechanics. Let us consider a set of N classicalparticles qj ,pjN with generalised coordinates qj and generalised momentapj , which obey the symplectic Hamilton equations

qj =∂HN

∂pj

pj = −∂HN

∂qj(3.1)

where HN ≡ H(qj ,pjN ) is the (time-independent) Hamilton functionof the system of classical particles [93]. Thus, we can formulate a Liouvilleequation by introducing a particle density function gN (qj ,pjN , t) [93]:

17

3. The Lattice Boltzmann Method (LBM)

∂gN∂t

= −N∑j=1

(∂gN∂pj

pj +∂gN∂qj

qj

)(3.2)

Although the Liouville equation provides a time evolution for the particledistribution function gN , we would like to have a simpler expression, ideallyinvolving a single particle distribution function g(qj ,pj , t) for particle j. Inorder to derive a simpler equation we need to employ three approximationsgiven in [93]:

1. only pairwise collisions of particles are considered (e.g. particle i withparticle j)

2. the pre-collision velocities of colliding particles are uncorrelated

3. particle interactions are not influenced by external forces

Now we can derive an evolution equation for the single particle distribu-tion function g(x,v, t):

∂g(x,v, t)

∂t+ v · ∇g(x,v, t) = Ω(g(x,v, t), geq(x,v, t)) (3.3)

which is the Boltzmann equation without external forces, where x = qjand v =

pjm are the position and velocity of the single particle j respec-

tively. m is the particle mass and Ω(g(x,v, t)) is the collision operator,which depends on the single particle distribution function g(x,v, t) and anequilibrium distribution function geq(x,v, t). The equilibrium distributionfunction can be derived by looking at the stationary state of equation (3.3)Ω(g(x,v, t), geq(x,v, t)) = 0 and we find

geq(x,v, t) = n

(m

2πkBT

) 32

e− m

2πkBT(v−v)2

(3.4)

which is the Maxwell Boltzmann distribution function with velocity field v,temperature T (x), total particle density n(x) and mean velocity v(x), wherekB is the Boltzmann constant [93]. We can now discretise the Boltzmannequation in both space and time on a lattice and obtain [94,95]:

gi(x + ci∆t, t+ ∆t)− gi(x, t)︸︷︷︸Streaming

= Ωi(gi(x, t))︸︷︷︸Collision

(3.5)

18


which is the Lattice Boltzmann equation (LBE). x are the discrete latticepositions and ∆t is the discretised time step. The index i now denotesa lattice stencil for each lattice position x with a set of discrete latticevelocities (or lattice vectors) ci, for which i ∈ 0 . . . N with N being thenumber of non-zero lattice vectors and ci=0 = 0 indicating the lattice nodeat position x. An example of such a lattice stencil is shown in figure 3.1for a D3Q19 lattice (3D and 19 lattice velocities ci). We remark that thediscretised single particle distribution functions gi(x, t) are often calledlattice populations or simply populations.

Figure 3.1.: Sketch of a 3DQ19 lattice with the discrete lattice velocities ci given aslattice directions.

The LBM equivalent of the equilibrium distribution function in equa-tion (3.4) is given by [93,94]:

geqi (ρ,v) = wiρ

(1 + 3(ci · v) +

9

2(ci · v)2 − 3

2v2

)(3.6)

with ρ(x, t) being the density of the system, v being the equilibrium veloc-ity and wi lattice weights [94]. The LBM equilibrium distribution functiongeqi is essentially a velocity truncated discretised Maxwell Boltzmann dis-

tribution function, see equation (3.4). The velocity truncation is especiallynecessary in the case of low compressibility, since in this region the Machnumber

Ma =v

cs(3.7)

19


should obey the limit Ma 1. v = ‖v(x, t)‖∞, with ‖. . .‖∞ being themaximum norm, and cs denotes the speed of sound in lattice Boltzmannunits (lbu), which for a uniform lattice is cs = 1√

3. We have indicated two

qualitatively different steps in equation (3.5), collision and streaming. Thecollision step merely evaluates the collision operator function Ωi(gi(x, t).The streaming step updates the values of the populations gi(x + ci∆t, t+∆t) at the neighbouring lattice nodes of node x with the post collisionalpopulations g′i(x, t+∆t). This provides us with the essential LBM algorithm:

Figure 3.2.: Sketch of the streaming step: The values of the populations gi(x+ci∆t, t+∆t)are updated with the post collisional population values g′i(x, t+ ∆t) at the lattice nodex.

1. perform the collision step via the collision operator Ωi(gi(x, t)

2. carry out the streaming step

3. evaluate and update the macroscopic quantities ρ(x, t) and v(x, t)

4. repeat step 1

The density ρ(x, t) and velocity field v(x, t) can be obtained via the LBMpopulations:

ρ(x, t) =∑i

gi(x, t)

v(x, t) =1

ρ(x, t)

∑i

gi(x, t)ci (3.8)

20


Ωi(gi(x, t) is now a generic discretised collision operator depending onthe whole set of lattice distribution functions gi(x, t) at lattice positionx and time t. It can be shown [95] that any generic LBM collision operatoronly depends linearly on the local populations gi(x, t) and the equilibriumdistribution functions geq

i (ρ,v):

Ωi(gi(x, t)) = Lij(gj(x, t)− geqj (x, t)) = Lijgneq

j (x, t) (3.9)

where gneqi (x, t) = gi(x, t) − geq

i (x, t) is the non-equilibrium part of thedistribution function gi and Lij the distribution function independent col-lision operator. The simplest collision operator is the BGK operator, forwhich Lij is diagonal and equal to a constant:

Lii =1

τ(3.10)

with τ being called the relaxation time of the system. According to equa-tion (3.9), τ indicates the speed at which the collision step takes place, i.e. atwhich rate the populations gi approach the equilibrium populations geq

i (x, t)or equivalently at which rate the non-equilibrium parts of the populationsgneqi (x, t)→ 0. Consequently the BGK operator models the collision with a

single relaxation time scale. For further numerical stability [96] we can usethe multi relaxation time scheme (MRT) [97]. In MRT we perform a basistransformation into a mode space with a set of orthonormal basis vectorseiji and corresponding modes mi(x, t):

mi(x, t) =∑j

eijgj(x, t) (3.11)

The collision step in mode space with relaxation parameters λi is thengiven via

m∗neqi (x, t) = (1 + λi)m

neqi (x, t) (3.12)

wherem∗neqi (x, t) is the post-collision non-equilibrium mode, andmneq

i (x, t)the pre-collision non-equilibrium mode. Important conserved modes are

• m0(x, t) = ρ(x, t)

• m1(x, t) = jx(x, t)∆t

• m2(x, t) = jy(x, t)∆t

21


• m3(x, t) = jz(x, t)∆t

which represent the mass ρ(x, t) and momentum density j(x, t) = ρ(x, t)v(x, t)respectively. For MRT the collision operator contains several relaxationtimes linked to its relaxation modes (depending on the lattice stencil) [97].In case of a fluid flow, one relaxation time τ is directly linked to the kine-matic viscosity ν in the system (incidentally it is equivalent to the one inthe BGK collision model)

ν = c2s

(τ − 1

2

)=

1

3

(τ − 1

2

)(3.13)

which is one of the primary links between the LBM scheme and hydro-dynamics [75, 76]. In this work we use the MRT collision model for allsimulations.

3.2. The Chapman-Enskog Expansion (CEE)

So far we have only dealt with LBM as a solver for the dynamics of particles,but have not seen their applicability to the continuum equations of fluidflows (equations (2.1)). The hydrodynamical manifold, i.e. the Navier-Stokesequations (NSE), can be recovered via the Chapman-Enskog expansion, aperturbation theory in the Knudsen number Kn, which is defined as

ε = Kn =lmL

(3.14)

where lm is the molecular mean free path length and L a macroscopiclength scale. The Knudsen number Kn can therefore be seen as an indicatorfor the physical density ρ of a system, e.g. for a fixed macroscopic length scaleL the Knudsen number Kn is directly proportional to the mean free pathlm, which decreases with increasing density. Now we define three relevanttime scales and one length scale for our perturbative expansion in Kn. Wewill be following the outline given in [95].

∆t ∼ O(1) lattice time scale

t1 = ε∆t ∼ O(ε) convective time scale

t2 = ε2∆t ∼ O(ε2) diffusive time scale

x1 = εx ∼ O(ε) coarse grained position vector (3.15)

22

3.3. Shan-Chen multicomponent model

With these scales defined we obtain a series expansion of equation (3.5)in orders of ε

O(1) : Ω(0)i = 0

O(ε) : (∂t1 + ci · ∇x1)g(0)i =

Ω(1)i

∆t

O(ε2) :[∂t2 + (∂t1 + ci · ∇x1)2

]g

(0)i +

∆t

2(∂t1 + ci · ∇x1)g

(1)i =

Ω(2)i

∆t(3.16)

with the indices (0), (1) and (2) denoting contributions of the order ofO(1), O(ε) and O(ε2) respectively. Thus, for the expansion of the LBE untilO(ε2) we obtain

∂tρ+∇ · (ρv) = 0

∂t(ρv) +∇ · (ρvv) = −∇p+∇ · σS (3.17)

with ∇x1 representing ∇ in respect to x1, the Euler stress Seqij = pδij +

ρ vivj and the non-equilibrium stress tensor S∗neq + Sneq = −2σ(S), where

σ(S)ij = ηijkl

∂vk∂rl

is the viscous stress tensor and ηijkl the dynamic viscositytensor. The pressure p(x, t) is given via an LBM equation of state

p(x, t) = ρ(x, t)c2s (3.18)

which can be seen as a pressure-density equivalency in the LBM frame-work. Equations (3.17) are the compressible Navier-Stokes equations (2.1)with a generic dynamic viscosity tensor, which reduce to the incompressibleNavier-Stokes equations in the incompressible (ρ(x, t) = ρ = constant) andthe Newtonian (η = ρν = constant) limits, where the dynamic viscosity ηis a scalar quantity.

3.3. Shan-Chen multicomponent model

The classical Lattice Boltzmann Model (LBM) for single phase flows needsto be modified to account for a system containing two immiscible fluids,in particular the fluid-fluid interface between them. One of the most used

23


scheme to model the fluid-fluid interface is the Shan-Chen Multi-Componentmodel (SCMC) [77, 78]. For two (or more) immiscible fluids we need todistinguish between the type of fluid component at hand, thus we get forthe mass and momentum densities:

ρ(x, t) =∑σ

∑i

gσi (x, t)

ρ(x, t)v(x, t) =∑σ

∑i

gσi (x, t)ci (3.19)

where gσi (x, t) denotes the populations in the LBM model for the fluidcomponent σ and ci are the lattice velocities. The interaction at the respec-tive fluid-fluid interface [98,99] is given by:

F σ(x) = −ψσ(x)∑σ′ 6=σ

N∑i=1

Gσ,σ′wiψσ′(x + ci)ci (3.20)

where ψσ(x) is a local pseudo-potential which may be defined via thephase densities ρσ(x, t). Gσ,σ′ is a coupling constant for the two phases σ andσ′ at position x and wi are the lattice isotropy weights. One should note thatthe stencil for the SCMC pseudo-potential interaction does not necessarilyhave to coincide with the stencil populations for the LBM streaming, butcould be a different lattice stencil, given that the interaction force F σ(x)remains isotropic.

3.4. Validation tests for confined droplets

After having introduced the LBM as a fluid solver for both single phase andmulticomponent flows, we give a LBM multicomponent flow example. Weconsider the static deformation of a 2D droplet centred in the middle of thesimulation domain, with periodic boundary conditions [94] in the x- andbounce-back (wall) boundary conditions [94] in the z-direction, see figure 3.3.The aim of the LBM simulations is to determine the deformation of the2D droplet as a function of the capillary number Ca under the influence ofan external shear flow ux at the boundary walls. ux at the boundaries is afree parameter in our simulations and ranges from the static case ux = 0 toux = 0.05 (lbu), which enables us to tune the capillary number Ca. In order

24


to determine the surface tension σ we carry out a Laplace measurement inthe static case, before starting the dynamic simulations involving ux. TheLaplace pressure in 2D is:

∆p =σ

R(3.21)

with ∆p ≡ |pin − pout| being the difference between the internal pressure(inner pressure) of the droplet and the external pressure of the surroundingfluid (outer pressure). The Laplace measurement is visualised in figure 3.4from which we can determine the surface tension σ due to (3.21). Forthe dynamic simulations we use a constant shear flow ux at the boundarywalls in the z-direction, see figure 3.3. In order to measure the deformationparameter D, we calculate the moment of inertia tensor Iij of the dropletat first:

Iij(t) =

∫VD

d3x ρ(x, t)(δij‖x‖2 − xixj

)(3.22)

where VD is the volume of the droplet [92]. Then we determine the majorL and minor axis W of the deformed droplet via the relation

L ∼ 1√µ2

W ∼ 1√µ1

(3.23)

with µ1 ≥ µ2 > 0, where µ1 and µ2 are the eigenvalues of Iij . This yieldsthe degree of deformation D defined in (2.8). Furthermore, for a comparisonof our simulations with the Shapira-Haber model [39,40] we need the shapefactor, which is Cs ≈ 5.6996 for a centred droplet. We can see in figure 3.5that our LBM simulations are in relatively good agreement with the Shapira-Haber model in the case of low capillary numbers Ca. The simulation resultsdeviate significantly from the theoretical model, however, if we consider arange of larger Ca values. This is to be expected, since the Shapira-Habermodel considers only the linear term in Ca of a perturbation series of p andv, see equations (2.6) 1.

1The disagreement between LBM and the SH-model vanishes, if we reduce the inertia,i.e. lower Reynolds numbers Re, as will be shown in chapter 4

25


2RLz

ux

-ux

x

z

Figure 3.3.: Set up for the Laplace test and static droplet deformation of a 2D confineddroplet in a shear flow. The droplet radius is given by R and the wall separation lengthby Lz. For the Laplace test the wall velocity ux = 0 and ux 6= 0 in the case of staticdeformation. The droplet is visualised in 3D for simplicity, even though the simulationswere carried out in 2D.

0.0028

0.00285

0.0029

0.00295

0.003

0.00305

0.0031

0.00315

0.0032

0.00325

0.0033

0.00335

0.027 0.028 0.029 0.03 0.031

Pin

-Pou

t

1/R

LBMBest Fit

Figure 3.4.: Laplace test ∆p against the interfacial curvature 1R

. The surface tension σis given via the slope of the best fit through the LBM data points.

26


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.02 0.04 0.06 0.08 0.1

Def

orm

atio

n

Ca

LBMBest Fit

SH

Figure 3.5.: Static droplet deformation test against the Shapira-Haber model (SH-model) [40] for a low Ca range. The best fit of the data points and the SH-model predictionsagree well in this capillary number range.

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

0 0.05 0.1 0.15 0.2

Def

orm

atio

n

Ca

LBMBest Fit

SH

Figure 3.6.: Static droplet deformation test against the SH-model [40] for a high Carange. The LBM data points do not follow a linear relationship and thus LBM disagreeswith the SH-model in this capillary number range. This disagreement is due to inertiaand vanishes for lower Reynolds numbers Re for the Ca range used here, see chapter 4.

27


3.5. A ghost node boundary flow scheme

In order to effectively simulate a time-dependent flow, we use speciallymodified boundary conditions. We make use of the ghost populations (orhalos) to store the local LBM equilibrium population distributions given bythe systems boundary values for the density ρb(x, t) and velocity u(x, t) ofthe outer fluid (for simplicity we will treat a single component fluid).

geqi (x, t) = ρb(x, t)wi

(1 + 3 ci · u +

9

2(ci · u)2 − 3

2u2

)(3.24)

with wi being the lattice weights for the set of lattice vectors ci.Thus the ghost distributions will update the boundary nodes during theLBM streaming step and let the system know about the previously chosenboundary conditions (see figure 3.7). Since the ghost nodes only streaminto the system and not out of it, we have to correct the local populationmass densities in order to keep the system mass conserving [11, 100–102],in the case of a confined system with a no slip boundary wall. Thus, theboundary scheme is equivalent to a mid-bounce-back rule [93, 94], if con-finement is present. However, if we deal with an unconfined system, whichwe would have in the case of a sub-Kolmogorov droplet in a turbulent flow,mass conservation does not have to be strictly enforced and we use an antievaporation scheme [11] to keep the mass of the droplet constant. This isa suitable procedure, since we are only interested in the dynamics of thedroplet and not that of the solvent. In this way we keep a constant densityratio and keep the droplet’s mass fixed, even though the mass of the solventmay vary.

28

3.5. A ghost node boundary flow scheme

Ghost nodes Boundary nodes

gieq gi

Figure 3.7.: Sketch of the streaming step from ghost to boundary nodes. The ghostnodes are initialised via the local equilibrium distributions geqi . By initialising the ghostnodes with a given equilibrium density ρ(x, t) and equilibrium velocity field u(x, t), wecan effectively set the boundary conditions of the system.

29

4. Droplet in a time-dependent flow:LBM and MM comparison

This chapter 1 deals with the ghost node boundary flow method outlinedin chapter 3. The method is first tested for a simple single phase problem.An oscillatory shear flow in an infinitely long channel. Then we use thisset up to to investigate droplet deformation in confined systems with bothstationary and time-dependent flows. The results of the LBM simulationsare compared to the ones obtained via numerical and perturbative solutionsto the MM-model [16,41].

4.1. Introduction

The multiscale problem of interest in this chapter is the fluid dynamics of anemulsion, i.e. a collection of small deformable droplets dispersed in a solventimmiscibile fluid: droplets can deform under the action of an imposed flowand can interact with neighboring droplets, they provide a back-reactionon the solvent component and ultimately determine the complex flowingproperties of the emulsion at large scales. Far from being only an interest-ing multiscale physical problem, it also finds a variety of applications inindustrial and engineering processes [1]. In order to describe these complexmulticomponent flows, we reduce the system to the dynamics and defor-mation of a single droplet under the influence of an externally imposedflow. The literature on single droplet deformation is vast, especially whendealing with laminar flows: following the pioneering work by Taylor [20],the deformation properties of a droplet have been extensively studied andreviewed [25, 103]. Existing studies address the effects introduced by thenature of the flow [104–107], the effects of confinement [40, 41], as well as

1Published as: Milan, F., Sbragaglia, M., Biferale, L., Toschi, F., Lattice Boltzmannsimulations of droplet dynamics in time-dependent flows, Eur. Phys. J. E (2018) 41: 6.https://doi.org/10.1140/epje/i2018-11613-0

31

4. Droplet in a time-dependent flow: LBM and MM comparison

the effects introduced by the complex non-Newtonian nature of the bulkfluids [87, 108, 109]. Exact analytical approaches are typically limited to“small” deformation assumptions, i.e. perturbative results. Extensions totime-dependent laminar flows have also been carried out [21]. From thetheoretical side, a popular model has been developed by Minale & Maffet-tone [16], the MM-model, which assumes the droplet to be an ellipsoid atall times [110,111], and is constructed to recover the perturbative results ondroplet deformation at small Ca (e.g. Taylor’s result [20]) in the presence of asteady flow. The MM-model has the key advantages to allow for time dynam-ics and also to extend the description of droplet deformation beyond the lim-its of applicability of perturbation theories [20], hence it has also been used tocharacterize the critical Ca for which droplet breakup occurs [16]. Followingthe MM-model, a whole class of “ellipsoidal” models have been introducedwith further enrichments to account for a variety of other effects, includingviscoselasticity [61, 112–114], confinement [29, 31–34,87, 88, 115] and match-ing with more refined perturbative results at small Ca [105,108,109,116]. Adetailed review on the topic can be found in [42]. These ellipsoidal modelsbecome particularly useful when studying the properties of a single dropletunder the influence of turbulent fluctuations [13,15,17]. Depending on thecharacteristic size of the droplet, turbulent fluctuations can provide eitherinertial distortions [13], when the droplet size is above the characteristicdissipative scale, or laminar distortions [15,17], for smaller droplets. It hasto be noted that analytical models cannot be used to describe the defor-mation of large droplets accurately and in particular the MM-model failsto capture non-ellipsoidal deformations. Therefore, it is crucial to developab initio models, such as multicomponent LBM with appropriate boundaryschemes in order to enforce time-dependent fluid deformations. If combinedwith a Lagrangian history of a turbulent strain matrix, the model allowsfor a comprehensive characterization of the statistics of droplet shape, sizeand orientation in a realistic turbulent environment [17]. A key parameterto quantify the reaction of the droplet to the time-dependent signal is theratio between the droplet relaxation time

td =ηdR

σ(4.1)

and the fluid time scale

32

4.2. Static droplet deformation

tf =R

u0(4.2)

where u0 is the maximal shear flow intensity. Depending on the ratio td/tfthe droplet is either able to follow the fluid variations (td/tf → 0), or startsto decouple when td/tf ≈ 1. This influences its deformation and possibly theallignment with the flow. Furthermore, a turbulent signal has a broad spec-trum rather than a single time scale tf, thus resulting in a multi-chromaticbehaviour coupled to the non-linear response of the droplet deformationprocess. This is an ideal workspace for LBM mesoscopic models to operate:they intrinsically allow for both droplet deformation at the mesoscale, andthey can be constructed to reproduce the desired hydrodynamical flow atlarge scales. Droplet deformation properties have been the subject of variousarticles [12, 29, 41, 43, 44, 48, 85, 103, 117–119], but these typically containstudies of deformation and orientation in steady state flows [12,80, 85, 119],or studies of the critical droplet breakup condition [46,88], with particularemphasis on the comparison between the (diffuse interface) hydrodynamicsof LBM and the sharp interface results [12, 87, 88, 119]. Droplet dynam-ics has also been simulated [25, 105, 108, 109, 111, 120], but the associatedquantitative validation has been scarcely detailed in the literature. Thischapter aims at filling this gap from the methodological point of view: afterrevisiting the validation of LBM for steady state flows, we introduce a timedependency to the system and quantitatively compare LBM against theanalytical predictions of ellipsoidal models at changing the ratio betweenthe droplet relaxation time td and the fluid time scale tf. The chapter isorganized as follows: section 4.2 gives a brief overview on the static dropletdeformation of Lattice Boltzmann simulations compared with relevant the-oretical models. In section 4.3 the behaviour of a simple oscillatory shearin a 2D LBM channel flow is tested, which will be relevant for section 4.4where we investigate the response of an isolated droplet to a time-dependentoscillatory channel flow in a 3D LBM model.

4.2. Static droplet deformation

The first step is to test our algorithm in the case of the deformation of asingle droplet in a constant shear flow confined in a channel, see figure 4.1.Inertia is characterised by the Reynolds number Re = R2G/ν where, in the

33


case of a simple shear flow, the shear rate is G = 2u0/Lz, with u0 beingthe maximum shear at the wall and Lz the channel width. Now we let thedroplet evolve in the shear flow and measure its deformation. We consideronly set-ups with an aspect ratio α ≡ 2R/Lz = 0.75 and keep the viscosityratio χ = 1 throughout all simulations. According to [121] the droplet will bestable up to a value of Ca ≈ 0.4 regardless of our choice for the confinementratio α. A series of LBM runs is shown in figure 4.2 and three different valuesfor the kinematic viscosity ν in lbu (Lattice Boltzmann units). We may seethat for the lowest value of ν the deformation D is deviating substantiallyfrom the theoretical predictions for a confined droplet, given both by theShapira-Haber model [29] and the MM-confined model [41].

Figure 4.1.: Screenshot of a multicomponent LBM simulation. A droplet is ellipsoidallydeformed via an external shear flow, created by the moving channel walls. The relevantsystem parameters are: the initial droplet radius R, the shear rate G, the channel width Lzand the other lengths of the simulation domain Lx and Ly (not shown). The magnitudeof the overall velocity field in lbu is given via a colour gradient.

For the two lower Re values the simulations agree much better with theMM-confined model predictions. Figure 4.2 also shows the static defor-mations for a higher resolution in the bottom panel where the respectiveReynolds number Re lies in the same range as for the plots in the toppanel. In this case the droplet deformation of our LBM scheme agrees evenbetter with the theoretical predictions for a confined droplet. Thus we candeduce both: that we need a significantly low Reynolds number and that we

34

4.3. Probing the parameter space: single component oscillating shear flow

may only compare our simulation results to models which account for theconfinement of the droplet. To our knowledge this is the first benchmark ofLBM against theoretical predictions for the influence of droplet inertia instatic droplet deformation in a system with a significant confinement ratio.

4.3. Probing the parameter space: single componentoscillating shear flow

After having benchmarked the static droplet deformation against a varietyof theoretical models we investigate the

droplet behaviour under a time-dependent linear shear flow. Before con-sidering explicitly the case of a binary fluid (see section 4.4) we need todetermine a suitable range for our LBM parameters. We remark that LBMworks well as a “hydrodynamic solver” only if the LBM populations areclose to the hydrodynamical manifold. Hence it is crucial to design a set of“working parameters” for which we know that our LBM scheme correctlysolves the time-dependent hydrodynamical equations. Specifically, for thecase of a time-dependent shear flow, we will compare our LBM schemeagainst the exact time-dependent solution of an oscillating shear flow [91].For simplicity we modify the boundary conditions for a 2D channel flow bysetting vx(0, t) = u0 cos(ωt) and vx(Lz, t) = 0, i.e. one side of the channel isoscillating with a shear frequency ωf = ω/(2π) and the other one is static(see figure 4.3). Making use of the incompressibility condition ∇ · v = 0 weobtain for the Navier-Stokes equation:

∂t vx = ν ∂2z vx. (4.3)

Making the ansatz

vx(z, t) = e−iωt (A cos(kz) +B sin(kz)) (4.4)

leads to the dispersion relation

k =1 + i

δ, (4.5)

where δ ≡√

2 νω is the penetration depth of the system. The solution for vx

reads

vx(z, t) = u0 e−iωt sin(k(Lz − z))

sin(kLz), (4.6)

35


Lower Resolution

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

D

Ca [2R/Lz=0.75]

TaylorShapira-HaberMM unboundedMM confinedLBM ν=1/6 (lbu)LBM ν=1/2 (lbu)LBM ν=5/2 (lbu)

Higher Resolution

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

D

Ca [2R/Lz=0.75]

TaylorShapira-HaberMM unboundedMM confinedLBM ν=1/3 (lbu)LBM ν=1 (lbu)LBM ν=7/6 (lbu)

Figure 4.2.: Droplet deformation test benchmarked against several theoretical models:perturbative models in the capillary number Ca, Taylor (unconfined droplet) [19,20] andShapira-Haber (accounting for droplet confinement) [39, 40] and models extending theCa range to higher values, MM-unbounded [16] and the MM-confined [41] model. Theextent of the deformation measured by the parameter D is plotted against the capillarynumber Ca. We choose a series of three Reynolds number ranges by selecting threekinematic viscosities. We can see that the agreement between the LBM simulation andthe theoretical predictions improves significantly for lower Reynolds numbers. This is dueto the reduction of inertia in the system. Top panel: Resolution 80 × 40 × 40 Bottompanel: Resolution 160 × 80 × 80. To account for similar Reynolds number ranges at ahigher resolution the simulations in the bottom panel have higher viscosity values thanthe one in the top panel.

36

4.3. Probing the parameter space: single component oscillating shear flow

vx(0,t) = u0cos(ωt)

vx(Lz,t) = 0

z

x

Lzvx,nonlin

vx,lin

Figure 4.3.: Sketch of the single phase model set-up. A 2D channel of width Lz with onestationary and one oscillating wall. The flow in the x-direction is periodically extended.Two typical velocity profiles of the oscillating channel flow are shown, a linear one in redand a nonlinear one in blue.

whose real part is denoted by v(0)x (z, t). The velocity profile vx(z, t) has

a linear limit, which is given by the penetration depth δ and the channelwidth Lz. If Lz/δ 1 the condition for a linear profile is fulfilled, and weget

v(lin)x (z, t) = u0 cos(ωt)

(1− z

Lz

). (4.7)

Thus we can find an upper bound for the frequency

ωc ∼ν

L2z

(4.8)

so that Lz/δc ∼ 1, with δc being the critical penetration depth of the system.With the analytical solution at hand we can now test our LBM scheme foran external shear flow in a channel including an exact time dependence andperform a “scanning” of the parameter space. In the following we define twoerror functions based on an L2-norm. Deviations from the exact analyticalsolution v0

x(z, t) are given by

E(0)x =

[1

LxLz

∫ Lx

0dx

∫ Lz

0dz (vx(x, z, t)− v(0)

x (z, t))2

] 12

. (4.9)

37


-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10

vx/u

0

t/T [ωf/ωc = 10-2

]

Upper channelMiddle channelLower channel

10-5

10-4

10-3

10-2

10-1

100

101

102

0 1 2 3 4 5 6 7 8 9 10

Ex(0

) /u0

Ex(lin

) /u0

t/T [ωf/ωc = 10-2

]

Ex(0)

/u0

Ex(lin)

/u0

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

vx/u

0

z/Lz [ωf/ωc = 10-2

]

LBM TExact T

LBM T/2Exact T/2LBM T/5

Exact T/5LBM 3/5 T

Exact 3/5 T

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10

vx/u

0

t/T [ωf/ωc = 10-1

]


10-5

10-4

10-3

10-2

10-1

100

101

102

0 1 2 3 4 5 6 7 8 9 10

Ex(0

) /u0

Ex(lin

) /u0

t/T [ωf/ωc = 10-1

]

Ex(0)

/u0

Ex(lin)

/u0

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

vx/u

0

z/Lz [ωf/ωc = 10-1

]

LBM TExact T


Exact T/5LBM 3/5 T

Exact 3/5 T

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10

vx/u

0

t/T [ωf/ωc = 1]


10-5

10-4

10-3

10-2

10-1

100

101

102

0 1 2 3 4 5 6 7 8 9 10

Ex(0

) /u0

Ex(lin

) /u0

t/T [ωf/ωc = 1]

Ex(0)

/u0

Ex(lin)

/u0

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

vx/u

0

z/Lz [ωf/ωc = 1]

LBM TExact T


Exact T/5LBM 3/5 T

Exact 3/5 T

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10

vx/u

0

t/T [ωf/ωc = 10]


10-5

10-4

10-3

10-2

10-1

100

101

102

0 1 2 3 4 5 6 7 8 9 10

Ex(0

) /u0

Ex(lin

) /u0

t/T [ωf/ωc = 10]

Ex(0)

/u0

Ex(lin)

/u0

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

vx/u

0

z/Lz [ωf/ωc = 10]

LBM TExact T


Exact T/5LBM 3/5 T

Exact 3/5 T

Figure 4.4.: Summary of a LBM benchmark series (channel width Lz = 128, relaxationtime τ = 1, maximal shear velocity u0 = 10−3) for a single component in the presenceof an oscillating shear. Left panels: time evolution of the velocity field at different heightlocations: upper channel (z = 3Lz/4), middle channel (z = Lz/2) and lower channel(z = Lz/4). We see a transient for higher frequencies ωf/ωc ∼ 10, which is due to therelaxation of the velocity field v(x, z, t) to the analytical solution in equation (4.6) froma zero velocity initialisation state. Middle panels: normalized error with respect to theexact solution (equation (4.9)) and normalized error with respect to the linearised solution(equation (4.10)). Right panels: Velocity profile vx(z, t) as a function of the dimensionlesscross-flow coordinate z/Lz at different times t = T/5, T/2, 3T/5, T (in units of the shearperiod T = 1/ωf ). In all plots we non-dimensionalise the shear frequency ωf by thecritical frequency ωc. We may notice qualitatively that the velocity profile is becominggradually more nonlinear with increasing frequency ωf after passing the critical regionωf/ωc ≈ 0.1. This is in agreement with the system parameter scan shown in figure 4.5.

38

4.4. Multicomponent oscillating flow

Moreover we define an error function with respect to the linearised solutionv(lin)x

E(lin)x =

[1

LxLz

∫ Lx

0dx

∫ Lz

0dz (vx(x, z, t)− v(lin)

x (z, t))2

] 12

. (4.10)

We perform several simulations with different oscillation frequencies rang-ing from ωf = 10−7 to ωf = 10−4. We can estimate the critical frequencyωc ≈ 10−5 via equation (4.8), which is supported by figure 4.4, as the veloc-ity profile starts to become linear at around this frequency value. We canalso see that the error E(0)

x with respect to the exact solution is (almost)independent of the oscillation frequency ωf . In addition, we now like toknow whether our LBM simulations produce similar results for a choiceof different system parameters. A sample of the parameter space is shownin table 4.1. Figure 4.5 is key in understanding the validity of our LBMsimulations for the single phase oscillatory shear flow. We can see both thatthe exact analytical error E(0)

x is generally well behaved (fluctuations arounda mean value) for the entire frequency range and that the error to the lin-earised solution E(lin)

x is well behaved for a frequency range ωf/ωc ≤ 10−1

and is increasing for higher frequencies. Thus, we may determine a frequencythreshold of about ωf/ωc ≈ 10−1 for the linear shear regime for which ourLBM solution is both stable and linear.


After having investigated the parameter space for a single phase system, weadd the droplet. We follow the discussion in [73] yielding a perturbativesolution in Ca. Firstly we consider the MM equation in a different non-dimensionalised form with respect to equation (2.13):

dMij

dt= Ca(t) [f2(SikMkj +MikSkj) + ΩikMkj −MikΩkj ]

− f1

(Mij − 3

IIIMIIM

δij

), (4.11)

where the time t is given in units of the droplet relaxation time td. Itis important to note that Ca(t) is now time-dependent due to the time-dependent external shear flow [70–73]. Following the discussion in [73], we

39


10-3

10-2

10-1

10-4

10-3

10-2

10-1

100

101

102

Ex(0

) /u0

ωf / ωc

Lz=32, τ=1.0, u0=10-1

Lz=64, τ=1.0, u0=10-1

Lz=128, τ=1.0, u0=10-1

Lz=32, τ=0.7, u0=10-1

Lz=64, τ=0.7, u0=10-1

Lz=128, τ=0.7, u0=10-1

Lz=32, τ=0.55, u0=10-1

Lz=64, τ=0.55, u0=10-1

Lz=128, τ=0.55, u0=10-1

Lz=32, τ=1.0, u0=10-2

Lz=64, τ=1.0, u0=10-2

Lz=128, τ=1.0, u0=10-2

Lz=32, τ=0.7, u0=10-2

Lz=64, τ=0.7, u0=10-2

Lz=128, τ=0.7, u0=10-2

Lz=32, τ=0.55, u0=10-2

Lz=64, τ=0.55, u0=10-2

Lz=128, τ=0.55, u0=10-2

Lz=32, τ=1.0, u0=10-3

Lz=64, τ=1.0, u0=10-3

Lz=128, τ=1.0, u0=10-3

Lz=32, τ=0.7, u0=10-3

Lz=64, τ=0.7, u0=10-3

Lz=128, τ=0.7, u0=10-3

Lz=32, τ=0.55, u0=10-3

Lz=64, τ=0.55, u0=10-3

Lz=128, τ=0.55, u0=10-3

Lz=32, τ=1.0, u0=10-4

Lz=64, τ=1.0, u0=10-4

Lz=128, τ=1.0, u0=10-4

Lz=32, τ=0.7, u0=10-4

Lz=64, τ=0.7, u0=10-4

Lz=128, τ=0.7, u0=10-4

Lz=32, τ=0.55, u0=10-4

Lz=64, τ=0.55, u0=10-4

Lz=128, τ=0.55, u0=10-4

10-3

10-2

10-1

100

10-4 10-3 10-2 10-1 100 101 102

E x(li

n)/u

0

ωf / ωc

Linear Nonlinear

Figure 4.5.: Scatter plot of the averaged analytical error function E(0)x ≈ 1

2u0(E

(0)x,max +

E(0)x,min) and averaged linear error function E

(lin)x ≈ 1

2u0(E

(0)x,max + E

(0)x,min) as a function

of the renormalised frequencyωf

ωcwith ωc = ν

L2z

and ωf ≡ ω2π

. The analytical error E(0)x is

relatively well behaved with a mean value of around 10−2 when averaged over the wholefrequency range for all parameter set-ups. On the other hand the linear error E

(lin)x shows

a clear dependence on the normalised frequency ωf/ωc. The linear error is well behavedup until a value of ωf/ωc ∼ 10−1, where the shear velocity profile starts becoming non

linear, which is demonstrated by the drastic increase in E(lin)x . The two regions, linear and

nonlinear, are separated by a dashed vertical line at ωf/ωc = 10−1.

40


expand the morphology tensor Mij as a perturbation series in the capillarynumber Ca(t), see appendix B. Ignoring an initial transient, we end up withthe following first order solutions for the squared ellipsoidal axes

L2 = 1 + Camaxf2

(ωtd cos(ωtdt)− f1 sin(ωtdt)

f21 + ω2t2d

)+O(Ca2

max),

W 2 = 1− Camaxf2

(ωtd cos(ωtdt)− f1 sin(ωtdt)

f21 + ω2t2d

)+O(Ca2

max),

B2 = 1 +O(Ca2max), (4.12)

where Camax denotes the maximal capillary number and t is given inunits of td. The quantities L2, B2 and W 2 denote the maximal, mediumand minimal eigen-directions of the morphology tensor Mij at all times tand are defined via

L2 = ||L2, W 2||∞,W 2 = 2− ||L2, W 2||∞,B2 = B2, (4.13)

where ||a, b||∞ ≡ max(|a|, |b|) is the maximum norm between two scalarquantities a and b. Besides the three ellipsoidal axes L, B and W anotherquantity is of particular interest to us. Analogously to [73] we applieda sinusoidal shear rate, so that for the time-dependent capillary numberCa(t) ∼ sin(ωtdt) (t in units of td). Thus we can identify a phase shift φbetween the external oscillatory shear and the droplet’s response given bythe time evolution of the squared ellipsoidal axes in equation (4.12):

φ = arctan

(ωtdf1

)+O(Ca2

max), (4.14)

which is (for the linearised solution) independent of Camax. With ourtheoretical model at hand we can now run LBM simulations of the dropletin the oscillatory shear flow and check the agreement with the perturbative

41


1

1.1

1.2

1.3

1.4

1.5

8 8.5 9 9.5 10

L/R

t/T [Camax = 0.12, Remax = 0.14 ωf td = 10

-3, ωf/ωc = 0.019]

LBMMM-unboundedMM-confined

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

8 8.5 9 9.5 10

B/R


-3, ωf/ωc = 0.019]


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

8 8.5 9 9.5 10

W/R


-3, ωf/ωc = 0.019]


1

1.1

1.2

1.3

1.4

1.5

8 8.5 9 9.5 10

L/R


-2, ωf/ωc = 0.19]


0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

8 8.5 9 9.5 10

B/R


-2, ωf/ωc = 0.19]


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

8 8.5 9 9.5 10

W/R


-2, ωf/ωc = 0.19]


1

1.1

1.2

1.3

1.4

1.5

8 8.5 9 9.5 10

L/R


-1, ωf/ωc = 0.96]


0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

8 8.5 9 9.5 10

B/R


-1, ωf/ωc = 0.96]


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

8 8.5 9 9.5 10

W/R


-1, ωf/ωc = 0.96]


Figure 4.6.: Numerical benchmark for LBM against the MM-confined solution of equa-tion (4.11). The time t is given in units of the shear period T = 1/ωf . W and L denote theminor and major axes respectively with B being the vorticity axis, where the ellipsoidalaxes obey L > B > W at maximal deformation. Four system parameters are of particularrelevance: Camax and Remax denote the maximal capillary and Reynolds number (givenfor the maximum shear at the channel walls) which remain fixed in the plots. ωf/ωc isa measure of the linearity of the shear flow, where ωf/ωc ∼ 1 may be seen as a limitingvalue for linearity (see section 4.3). ωf td denotes the oscillation frequency in units of thereciprocal droplet relaxation time and is the control parameter here. We may observequalitatively that as the oscillation frequency tends to values close to ωf td ∼ 1 the dropletdeformation decreases and undergoes a phase shift with respect to the outer shear flowat the walls.

42


theoretical predictions in equation (4.12). However, the perturbative an-alytical solution is only valid in a small capillary Camax range. We can alsosolve the time-dependent MM-confined equation (4.11) and compare thenumerical solution (obtained via a RK-4 scheme) to our LBM simulationresults instead of the perturbative solution. It should be remarked thatour LBM simulation results may only be compared to the MM-confinedmodel, when the droplet remains an ellipsoid at all times. In order to havea complete overview of the droplet deformation it is paramount to visualiseall three major ellipsoidal axes L, B, W , where L > B > W at maxi-mum deformation. Let us look at the top row of figure 4.6. Consideringthe time evolution of the major axis L and minor axis W we can see thatMM-unbounded model is not properly accounting for the confinement ofthe system (α = 0.75). On the other hand our LBM simulation results arein relatively good agreement with the numerical solution of MM-confined.Interestingly, the vorticity axis B is also deformed in time, which is notthe case in the perturbative model, since the deformation is due to higherorders O(Ca2

max) in this case. Moving one row further down in figure 4.6, i.e.increasing the previous frequency by a factor 10 we can observe two changes.Firstly, the value of the droplet deformation D is decreasing (for both LBMand the MM-confined solution), and secondly, the time evolution is shiftedwith respect to the previous row. These effects may be explained by thedroplet inertia which tries to resist the outer shear flow. Since ωf td ∼ 10−2

we are in the regime where the droplet relaxation time scale is relativelyclose to the oscillatory shear period 1/ωf . Therefore, we may expect botha decrease in deformation and a phase shift φ between the outer shear flowand the time-dependent droplet deformation D(t), as is also predicted bythe analytical perturbative solution (see equation (4.14)). The phase shiftφ is measured in the LBM simulations by the difference in simulation timebetween the maximal shear intensity G and the maximal droplet deforma-tion D. Increasing the frequency even more to ωf td ≈ 10−1 the deformationdecreases substantially and the phase shift φ is close to π/2. This indicatesthat as ωf td → 1 the droplet is behaving as if the flow was not present at all.We call this the “transparency” effect, since the droplet seems to be (almost)transparent to the surronding flow field, which makes itself noticeable bythe droplet’s out of phase response and drastic decrease in the deformationparameter D. This decrease in deformation due to a phase shift betweenapplied shear and droplet response has also been experimentally confirmed

43


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

D

Camax

ωf td=6x10-4

ωf td=1x10-3

ωf td=2x10-3

ωf td=2.5x10-3

ωf td=6x10-3

ωf td=1x10-2

ωf td=2x10-2

ωf td=2.5x10-2

ωf td=3x10-2

ωf td=4x10-2

ωf td=6x10-2

ωf td=1x10-1

Shapira-Haber (static)MM-confined (static)

Figure 4.7.: LBM obtained droplet deformation D against Camax for various valuesof the normalised shear frequency ωf td. The transparency effect, i.e. the reduction ofdeformation D for ωf td → 1 (see equation (4.12) and figure 4.6) is confirmed. For furtherclarification the Shapira-Haber and the MM-confined static deformation curves are shown,from which we may see that a relatively low oscillation frequency of about ωf td ∼ 10−3

results already in a noticeable decrease in D.

44


by Cavallo et al. [72], where the authors use a different small amplitudemodel as a benchmark for their experimental results. For further analysisfigure 4.7 shows the LBM droplet deformation results as a function of Camax

for the simulated frequency range. In figure 4.7 the transparency effect isshown in a more quantitative way. We observe for various simulations, thatthe deformation drops significantly for increasing frequency ωf , indepen-dently of the capillary number Ca. For further comparison of the dropletdeformation scale D the Shapira-Haber [39,40] and MM-confined curves [41]are given as well. Figure 4.8 shows both the deformation D and the phaseshift φ between the droplet response and the oscillatory shear flow as afunction of the normalised frequency ωf td. The general trend is that thedeformation D is stationary up until ωf td ∼ 0.01 at which point D startsdecreasing until the droplet becomes “transparent” to the outer shear flow.We may also see that the phase shift φ is starting to increase rapidly atωf td ∼ 0.01 from φ ∼ 0 up to φ ∼ π/2. This reinforces the idea of the droplettransparency effect. Similarly to a forced harmonic oscillator the shear flowis out of phase with the droplet’s response, because the oscillatory shearperiod 1/ωf is of comparable size to the droplet relaxation time td. Thenumerical solution of the MM-confined equation (4.11) shown in figure 4.8 isin good agreement with the perturbative analytical solution equation (4.12)for both D and φ. The LBM results predict a smaller deformation D anda larger phase shift φ compared to the MM-confined model. This may beexplained by the very thin droplet interface in the LBM simulation whichis roughly the size of 1 grid point. From this estimation we can deduce arelative error of about 0.02 to both the time-dependent values of L and W ,resulting in a relative error of about 0.01 for the deformation parameter D.Moreover, it is useful to qualitatively consider streamline plots of the dropletdynamics in both the low and high frequency regions (see figure 4.9). Inthe low frequency regime for ωf td = 0.001 in figure 4.9 we see the familiarcase of static droplet deformation [87], where we have a tilted ellipsoidallydeformed droplet (in agreement with the MM-confined model) in the case ofmaximum deformation coinciding with the instance of the maximum sheardue to φ 1. In the high frequency regime ωf td = 0.1 in figure 4.9 wesee now that the droplet is only slightly deformed in the case of maximaldeformation. Since the phase shift φ ≈ π/2 now, the velocity magnitude ofthe oscillatory shear flow is almost 0 at the walls. We observe that the re-gions of highest shear flow intensity are in fact close to the droplet interface

45


(disregarding the two channel wakes produced by the droplet). Thus theinternal droplet dynamics substantially influences the oscillatory shear flowclose to the interface in the high frequency regime. This is a consequenceof the two-way coupling of the Multicomponent LBM scheme.

0

0.05

0.1

0.15

0.2

0.25

0.001 0.01 0.1 0

10

20

30

40

50

60

70

80

90

D φωf td [Camax=0.12]

LBM DMM-confined solution D

MM-confined perturbative D

LBM φMM-confined solution φ

MM-confined perturbative φ

Figure 4.8.: Droplet deformation D and phase shift φ between the shear flow and thedroplet’s response for a fixed maximal capillary number Camax against the normalisedfrequency ωf td. The LBM measured droplet deformation D is in good agreement withthe MM-confined prediction (see figure 4.2 for the static benchmark). The droplet trans-parency effect seems to come into effect at around ωf td ∼ 10−2, where a gradual decreasein D is noticeable in both MM-confined and LBM results. MM-confined and LBM agreefurthermore on the phase shift φ which increases from φ ∼ 0 to φ ∼ π/2 for the highestmeasured frequencies. This indicates an out of phase droplet response to the underlyingshear flow. For further clarification the linearised perturbative MM-confined solution,equation (4.12), is also shown.

4.5. Conclusion

We have demonstrated that a Shan-Chen multicomponent LBM set up withparticularly chosen boundary conditions yields reliable results for confinedtime-dependent droplet deformation. After validations in the static case [19,20, 39, 40], we have checked the LBM results against a variety of time-dependent theoretical models [16, 41, 73]. Specifically, after introducing a

46

4.5. Conclusion

Low shear frequency ωftd = 0.001, Camax = 0.12

High shear frequency ωftd = 0.1, Camax = 0.12

Figure 4.9.: Streamline plots of the LBM droplet simulations at maximum deformationfor Camax = 0.12 and Remax ≈ 0.1. Top panel: Low frequency regime ωf td = 0.001. Thedroplet is ellipsoidally deformed and tilted in the channel, similarly to static dropletdeformation dynamics [87]. The droplet produces two wakes in the channel and thevelocity magnitude is largest at the channel walls. Bottom panel: High frequency regimeωf td = 0.1. The droplet is only marginally deformed due to the “transparency effect”at non-dimensionalised high frequencies ωf td. We observe once again two wakes in theflow field in the vicinity of the droplet. Due to φ ≈ π/2 (phase shift between underlyingoscillatory shear and droplet deformation) the velocity magnitude is largest at the dropletinterface instead of the channel walls. The droplet is deforming the underlying oscillatoryshear flow through its own internal dynamics (two-way coupling).

47


time dependence into the system via a monochromatic shear, the LBMsimulations agree fairly well with theoretical models and discrepancies arelikely due to the interface thickness in the LBM model. The simulationsin this work have been carried out with a boundary scheme using ghostnodes which is, on the one hand, equivalent to a wall bounce back scheme,but, on the other hand, may be extended to model more complex externalflows than those treated here. Therefore, our simulations, both in the caseof single pahse and multicomponent flows, are useful validation tests of aboundary method involving ghost nodes, which can be extended to an exactflow boundary scheme, similarly to [101, 102]. The work in this chapter isextended to consider the rather interesting aspect of frequency dependentdroplet breakup in highly confined flows in chapter 5. Furthermore, we willsee in chapters 6 and 7 that the underlying LBM ghost node boundary flowscheme described here may be extended to accurately simulate an “ab-initio”droplet in a turbulent flow [13,15,17].

48

4.5. Conclusion

Lz (lbu) τ (lbu) u0 (lbu) ωf (lbu) E(lin)

x,min/u0 E(0)x,max/u0

128 1.0 10−2 10−7 0.7461 · 10−2 0.2442 · 10−2

128 1.0 10−2 10−6 0.7527 · 10−2 0.6827 · 10−2

128 1.0 10−2 10−5 1.2556 · 10−2 1.1707 · 10−2

128 1.0 10−2 10−4 1.8881 · 10−2 9.9081 · 10−2

64 1.0 10−3 10−7 1.542 · 10−2 0.063 · 10−2

64 1.0 10−3 10−6 1.543 · 10−2 0.595 · 10−2

64 1.0 10−3 10−5 1.633 · 10−2 1.171 · 10−2

64 1.0 10−3 10−4 2.909 · 10−2 4.833 · 10−2

128 0.7 10−3 10−7 0.447 · 10−2 0.679 · 10−2

128 0.7 10−3 10−6 0.486 · 10−2 0.701 · 10−2

128 0.7 10−3 10−5 5.087 · 10−2 1.106 · 10−2

128 0.7 10−3 10−4 12.972 · 10−2 1.558 · 10−2

128 1.0 10−3 10−7 0.243 · 10−2 0.803 · 10−2

128 1.0 10−3 10−6 0.660 · 10−2 0.810 · 10−2

128 1.0 10−3 10−5 1.185 · 10−2 1.192 · 10−2

128 1.0 10−3 10−4 9.900 · 10−2 1.893 · 10−2

Table 4.1.: System parameter scan of the single phase oscillatory channel flow (seesection 4.3). A few representative cases are reported of changing the channel width Lz,the LBM relaxation time τ , the maximum wall velocity u0, the shear oscillation frequencyωf .

49

5. Droplet breakup in a confined andtime-dependent flow

In this chapter 1 we investigate droplet breakup in confined and time-dependent oscillatory shear flows using the same set up as in chapter 4.We see that breakup is not only dependent on the oscillatory frequency ofthe outer shear flow, but also on start up conditions and the degree of con-finement: two very different flow start up conditions yield two quantitativelydifferent results in the case of low shear frequencies and high degrees of con-finement. Moreover, we investigate the effect of inertia and flow topologyon the breakup conditions and the mismatch between the two flow start upprotocols.

5.1. Introduction

Fluid dynamics phenomena, involving droplet dynamics, deformation andbreakup, are prominent in the field of microfluidics and even in general com-plex flows at larger scales. Beyond the practical importance in a variety ofconcrete applications [1–4], they are also relevant from the theoretical pointof view, due to the complexity of the physics involved [19, 22, 25, 27, 103].The value of the capillary number Ca at breakup is denoted by Cacr, thecritical capillary number. A lot of attention has been dedicated to dropletdeformation and breakup in stationary flows [22–24], and in particular theeffect of the degree of confinement α, see equation (2.19), on the flow dynam-ics [26,28,29]. Confinement is frequently encountered in experimental set upsof droplet dynamics in simple shear flows [26,28–38] and can be enhancedby changing α. There are some theoretical models which were developedto capture the experimental phenomenology of confined droplet dynamics,analytical models [39,40], which extended the theory by Taylor [19,20], and

1Submitted as: Milan, F., Biferale, L., Sbragaglia, M., Toschi, F., Lattice Boltzmannsimulations of droplet breakup in confined and time-dependent flows

51

5. Droplet breakup in a confined time-dependent flow

phenomenological models [16,41,42]. The validity of the analytical modelswere verified in [43] and the phenomenological models in [44]. Of particularinterest are the results in [28], which show that, for non vanishing α breakupdiffers substantially from the unconfined shear case both qualitatively andquantitatively for various viscosity ratios χ. Additionally, the dependencyof the critical capillary number Cacr on the droplet’s inertia is a central areaof interest [12,25,45–64]. Furthermore, breakup is influenced by the start upconditions, as demonstrated in experimental and theoretical studies [65–69].This phenomenon is rather subtle and can have different effects dependingon the protocol in use. The dependency on the rate of increase of the shearrate G was confirmed by [65] via supporting calculations based on the modelby Taylor [19]. A theoretical model developed by Hinch et al. [66] shows thatstable droplet equilibria below the critical capillary number Cacr are onlypossible for a sufficiently low increase in G. Furthermore Renardy [69] hasshown that, although these stable equilibria require a slow increase in theshear rate G, they are unique and do not depend on the rate of change ofG. We stress that even though the effect of start up conditions on breakuphas been investigated [65–69], the role of confinement with varying start upconditions on droplet dynamics and breakup is not clear. Moreover, it isunclear how breakup is affected, if the flows are time-dependent [70–74]. Theaim of this chapter is to take a step further in this direction. With the use ofnumerical simulations we show that at capillary numbers close to breakup,confinement allows for the existence of a metastable flow configuration nextto the solution of the Stokes equation found in [69]. This metastable stateis prone to perturbations and collapses to the Stokes solution, if we havea time dependent flow with a sufficiently large shear frequency. It shouldbe stressed that this result is unique to the case of a confined droplet inan oscillatory shear, as this metastable configuration is not present neitherfor an unconfined droplet in an oscillatory shear flow nor in the case ofan oscillatory elongational flow. Our studies can be seen as an extensionto [46, 69], where the influence of inertia on droplet breakup was studied,whereas we deal with time-dependent cases, where the temporal rate ofchange of the shear intensity is comparable to the droplet relaxation time.This chapter extends the work produced in chapter 4, where stable time-dependent droplet dynamics was investigated via a multicomponent LatticeBoltzmann scheme [74] and a phenomenological model [16,41].

52

5.2. Simulation set up and definitions

This chapter is organized as follows: in section 5.2 we outline the generaldetails of droplet breakup with an emphasis on confined systems and simpleshear flows. In section 5.3 we investigate breakup in a time-dependent(oscillatory) shear flow under strong confinement. A mismatch between twoprotocols, involving different start up conditions of the flow, leads us toinvestigate breakup conditions under the influence of inertia (section 5.4)and the effect of confinement (section 5.5), Moreover, we check whether theprotocol mismatch depends on the flow topology (section 5.6).

5.2. Simulation set up and definitions

In this section we define what we mean when we speak of droplet breakupand characterise the simulation set ups. We deal with both a confineddroplet in a simple shear flow and an unconfined droplet in a uniaxialextensional (elongational) flow. The velocity gradient matrix for both shearand elongational flows is given by

∇v =G

2

β 0 2(1− β)0 β 00 0 −2β

(5.1)

where ‖∇v‖ = G and β is a parameter characterising the flow type.The shear flow set up is equivalent to the one used in [74] with β = 0 inequation (5.1) except that the flow is unconfined and elongational with anoscillatory velocity gradient amplitude G(t) given by equation (5.1) withβ = 1. Droplet deformation can be characterised by the capillary numberCa. In the case of a shear flow including confinement the shear rate is givenby

G =2u0

Lz(5.2)

with Lz being the channel width responsible for the droplet confinementand u0 being the maximum wall velocity amplitude. This definition mayalso be extended to time-dependent shear flows [74]

G(t) =2u(t)

Lz(5.3)

53


Ca < Cacr

t=t1 t=t2

t=t3t=t3

L(t)

t=t4

L(t)

t=t5

Lcr(t)

t=t7 t=t8

Ca > Cacr

t=t6t=t5

Lz

Lz

2R 2

3

4

5

6

7

8

200 400 600 800 1000 1200 1400

Dro

plet

leng

th L

(t)/R

t/td

t1t2

t3t4

2

3

4

5

6

7

8

50 100 150 200 250 300 350 400

Dro

plet

leng

th L

(t)/R

t/td

t5t6 t7

t8

Figure 5.1.: Droplet in a confined oscillatory shear flow with a non-dimensionalisedoscillation frequency ωf td. Snapshots of the droplet in the velocity field are shown forCa < Cacr and Ca > Cacr. The plots on the right panel shows the time evolution of thenormalised droplet length L(t)/R. The degree of confinement of the system is given byα = 2R/Lz, where R is the droplet radius of the undeformed droplet and Lz the wallseparation.

54

5.3. Droplet breakup in an oscillatory shear flow

In accordance with [28] we define the critical capillary number Cacr asthe value of Ca for which an initially spherical droplet breaks up, which isachieved by a sudden increase in the shear rate amplitude G. We refer tothis breakup protocol as the Shock Method. In addition we can graduallyincrease the shear rate G starting from a value for which the droplet isonly marginally deformed [65, 66, 69]. A fixed increase ∆G, or ∆u0 in thecase of equation (5.3), is equivalent to a fixed increment rate ∆Ca for thecapillary number. This way the droplet and the solvent flow are given moretime to relax to their respective equilibrium distributions at specific Ca.We call this protocol the Relaxation method. A variation of the relaxationmethod for time-dependent oscillatory flows, i.e. where the shear amplitudeG(t) = G0 cos(ωt), is to consider the flow and droplet configuration ata capillary number Ca close to Cacr and then to increase the oscillatoryshear frequency ωf = ω/(2π) until breakup, starting from the stationarycase of ωf = 0. As in [74] we use a dimensionless frequency ωf td in ourdiscussion, where td is the droplet relaxation time defined in equation (4.1).In the presence of a flow with non-zero frequency ωf td, we focus on Camax,which denotes the maximum value of the time-dependent capillary numberCa(t) over one oscillatory cycle [74]. An instance of droplet breakup inan oscillatory simple shear flow is depicted in figure 5.1. The droplet isoscillating between two maximally elongated states for Ca < Cacr andbreaks up during the flow build up for Ca > Cacr in the case of the shockmethod. The droplet elongation is characterised by the droplet length L(t),which is defined as the longest axis of the elongated droplet and Lcr denotesthe droplet length in the critical case Ca ≥ Cacr. The time evolution ofL(t) is also shown for the two cases Ca < Cacr and Ca > Cacr in figure 5.1,which shows that breakup occurs at around t = 17000 lbu with lbu denotingLattice Boltzmann Units. In all simulations in this article the viscous ratioχ ≡ 1 and the density ratio ρd/ρs ≡ 1. If not explicitly stated otherwise,the confinement ratio is set to α = 0.75.

5.3. Droplet breakup in an oscillatory shear flow

Similarly to [74] we consider a droplet in a confined oscillatory shear flow,see figure 5.2, for the flow pattern. The set up is shown in figure 5.1 witha confinement ratio α = 0.75 and a time-dependent shear rate G(t) =2u0/Lz cos(2πωf t), where ωf is the frequency of the outer oscillatory flow [70–

55


Figure 5.2.: Planar cut of a droplet in a shear flow, where the flow is visualised bystreamlines coloured according to the magnitude of the velocity field.

74]. Our main focus is the dependency of Cacr on the normalised shearfrequency ωf td of the oscillatory outer flow. Droplet dynamics in oscillatingshear flows feature a so called transparency effect [74], which states that thedroplet is hardly deformed, if ωf td ∼ 0.1, i.e. the time scale of the oscillatingshear flow 1/ωf is of a similar order as the droplet relaxation time scaletd. The droplet dynamics are hardly influenced by the shear frequency forωf td ∼ 10−4 and the transparency effect is noticeable for ωf td ∼ 10−2 andhigher frequencies, which leads to a sudden increase in the critical capillarynumber. To be able to compare the LBM simulations with experimentalresults [26, 28, 35, 36], we limit the range of the critical capillary numberclose to Cacr ∼ 1.0. In figure 5.3 we can see that the droplet breakupbehaviour is significantly different for our two LBM simulation protocols,the shock and relaxation method. The shock method implies that dropletbreakup is independent of the oscillatory shear frequency ωf td, significantchanges in Cacr only occur close to the transparency effect region at highfrequencies (ωf td ∼ 10−2). The relaxation method is of a different nature:first of all Cacr in the low frequency region (ωf td ∼ 10−4) is larger thanthe values obtained with the shock method, see also section 5.5. Moreover,for intermediate frequencies ωf td ∼ 5 × 10−3 we observe that breakupoccurs at a significantly smaller Cacr than in the low frequency range andis now of a comparable value to Cacr obtained via the shock method. Themismatch between the two protocols in the low frequency regime in figure 5.3is in disagreement with previous studies of start up conditions of droplet

56

5.4. Inertial effects

breakup in confined simple shear flows [28,69]. However, the shock methodproduces results in accordance with the literature [28], as the dashed linein figure 5.3 indicates. It should also be noted, that the destabilization ofthe “relaxation branch” is rather sudden and takes place at very small ωf td.This suggests that the protocol mismatch is due to a metastable solution(relaxation method) existing next to a stable solution (shock method) inthe low frequency range ωf td ≤ 0.02. The protocol mismatch seems ratherpuzzling: according to Renardy [69] the solution should be unique. However,our set up differs in a few points from the one in Renardy [69]. First of all,the droplet is strongly confined (α = 0.75) in our set up (see figure 5.1),which could have a strong effect on the values Cacr for varying start upconditions. Moreover, inertia might stabilise the droplet in the case of therelaxation method. Therefore, the protocol mismatch might disappear inthe Stokes limit. In addition, one may also wonder, if the flow topologyplays a role, as an inherently different flow field might lead to a similarprotocol mismatch. Given these considerations, we investigate the cause ofthe mismatch by considering both inertial effects, as is the case in [69], seesection 5.4, and the importance of confinement in stationary shear flows,see section 5.5, in the following sections. Regarding the importance of flowtopology, we investigate time-dependent breakup in an elongational flow insection 5.6.

5.4. Inertial effects

In [69] it is shown that the solution of the Stokes equation in confinedsimple shear flows is unique and does not depend on neither the initialconditions of the droplet nor the solvent flow configuration. Thus, one mightthink that the protcol mismatch might be due to inertial effects and woulddisappear, if we were close the Stokes limit of Re ≡ 0. Interestingly, theLBM formalism allows us to directly set Re = 0, as we can eliminate the non-linear terms in the equilibrium distribution functions in the LBM algorithm,equation (3.6), which leads us to a modified equation( 5.4), accountingonly for the linear terms in the flow field u(x, t). Inertial effects tend tostabilise the droplet [67,68] for low Re < 1, whereas Cacr ∼ 1/Re for largeRe > 10 [46]. This suggests, that the stabilisation effect of low Re areresponsible for the protocol mismatch, which consequently should disappearin the Stokes limit Re = 0. We investigate the dependency of Cacr on Re,

57


0.4

0.5

0.6

0.7

0.8

0.9

1

10-3

10-2

Ca

cr

ωf td

LBM: Shock methodLBM: Relaxation method

Cacr for ωf = 0 Jansen (2010) (Experiment)

Figure 5.3.: Critical capillary number Cacr at varying frequencies ωf td. There is amismatch between the predictions of the two breakup protocols. Whereas droplet breakupis largely independent in the case of the shock method, except for the asymptotic behaviourin the high frequency region, the relaxation method in the low frequency limit predictsa higher Cacr than the ones of the shock method. This mismatch is investigated in thearticle. The error bars are estimated via steps in the critical capillary number ∆Ca, whichindicate the steps taken during the flow start up in the relaxation method. Both curvesare interpolated via bezier curves.

58

5.5. Confinement effects

as shown in figure 5.4. For the case Re = 0 we use only the linear terms ofthe equilibrium distribution functions given by

geq,lini (x, t) = ρb(x, t)wi (1 + 3 ci · u) (5.4)

The simulations are carried out for a stationary shear flow, with the set updescribed in figure 5.1. We can see that the mismatch between the breakupprotocols does not depend on inertia and is even present in the Stokeslimit of Re = 0. We conclude that the mismatch between the two breakupprotocols is not influenced by any stabilisation effects of inertia [67, 68] forthe given range of Reynolds numbers Re ∼ 0.0, . . . , 1.5.

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ca

cr

Re

Shock method

Relaxation method

Figure 5.4.: Cacr vs Reynolds number Re. The mismatch between the shock and relax-ation breakup protocols does not depend on inertia. This is especially clear in the caseof the Stokes solution, for which Re = 0. The error bars are estimated via steps in thecritical capillary ∆Ca and Reynolds number ∆Re.


Now we focus on both confinement and start up conditions in the shear rateamplitude G for droplet breakup in a stationary shear flow. The set up isonce again the one in figure 5.1, a confined droplet in a stationary (ωf td = 0)shear flow, but now we vary the confinement ratio α and, in the case of therelaxation method, the rate of change of the shear amplitude G, resulting

59


in increments of the capillary number ∆Ca. Our results are summarisedin figure 5.5. We can see, as was shown in [69], that the critical capillarynumber Cacr is independent of the start up conditions for low confinementratios (α ≤ 0.5), as both the shock method and the relaxation methodyield the same results with respect to the simulation errors. However, if thedroplet is strongly confined (α ≥ 0.6), the two methods yield very differentresults, with the Cacr predicted by the relaxation method being substantiallylarger than the one predicted by the shock method. Figure 5.6 shows thelength of the elongated droplet as a function of the LBM simulation time forthe different shear start up methods: we can see that for the shock methoddroplet breakup occurs soon after the maximal elongation, whereas for therelaxation method the droplet experiences a sequence of maximal extensionsand subsequent retractions after breaking up, for a given Cacr at its criticallength Lcr(t). We conclude that both a slow start up of the outer flow(relaxation method) and a strong confinement of the droplet (α ≥ 0.6) arenecessary for the mismatch reported in figure 5.3 in the low frequency limit.The eventual collapse of the relaxation method solution on to the one foundby the shock method suggests, that the relaxation method branch in thelow frequency limit in figure 5.3 is a metastable state, explaining the highsusceptibility to small perturbations and the collapse to the configurationobtained by the shock method for intermediate oscillatory frequencies ωf td.

60


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1

Ca

cr

α

Shock method

Relaxation method, ∆Ca=0.04











Figure 5.5.: Critical capillary number Cacr for different confinement ratios α = 2R/Lz.We compare the values obtained by the LBM simulations with the shock method and theones obtained by the relaxation method. Since the relaxation method is dependent onthe start up conditions of the outer flow and the droplet, we provide a range of differentincrements ∆Ca, where smaller ∆Ca denote a slower and flow build up and vice versa.The error bars are estimated via steps in the critical capillary number ∆Ca.

61


0

5

10

15

20

0 200 400 600 800 1000 1200 1400 1600 1800

L(t

)/R

t/td

Shock method: Ca=0.54Relaxation method: ∆Ca=0.30Relaxation method: ∆Ca=0.24

Figure 5.6.: Normalised droplet major axis L(t)/R against the LBM simulation timet given in units of the droplet relaxation time td. The droplet breaks up shortly afterits maximum elongation for the shock method. breakup in the relaxation method isdependent on the shear rate and thus capillary number increase: a) for a rate withincrement ∆Ca = 0.30 the droplet relaxes after reaching its maximum elongation for thefirst time to breakup at a longer length at a higher Cacr later on. b) for a smaller capillarynumber increase ∆Ca = 0.24 the droplet length at Cacr increases even further and theL(t) contains more full extensions and subsequent retractions.

62

5.6. Flow topology

5.6. Flow topology

After having dealt with confinement effects, we investigate the protocolmismatch in terms of the flow topology. Instead of an oscillatory shear flow,we consider breakup in an elongational (or uniaxial extesnional) flow, seefigure 5.7. This flow is by its very nature unconfined, so we would expectnot to see a mismatch, as is the case for α = 0 in the case of the confinedshear flow, see section 5.5. The results are shown in figure 5.8. Interestingly,a mismatch between the two droplet protocols is absent and the predictionsagree well with each other in terms of their respective errors. This showsthat strong confinement (α ≥ 0.75) is necessary for the existence of theprotocol mismatch shown in figure 5.3. Moreover, figure 5.8, shows thatdroplet breakup in an oscillatory elongational flow is frequency dependent,with an exponential dependence between the oscillation frequency ωf td andthe critical capillary number Cacr. The low frequency limit matches thestationary flow predictions of [51].

Figure 5.7.: The flow layout of a droplet in an elongational (uniaxial extensional) flow.The image is a planar cut, with the flow being rotationally symmetric around the elongateddroplet axis in the image. The streamlines are coloured according to the magnitude ofthe velocity field of the solvent.

63


0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

10-4

10-3

10-2

Ca

cr

ωf td

Shock methodRelaxation methodCacr for ωf = 0 Feigl (2002) (BIM)

Figure 5.8.: Critical capillary number Cacr against different frequencies ωf td for a dropletin an unconfined elongational flow. We consider the two breakup protocols, the shockmethod and the relaxation method. Even though the droplet breakup is dependent onthe shear rate frequency ωf td, a protocol mismatch does not occur, contrary to the caseof the confined shear flow topology. The error bars are estimated via steps in the criticalcapillary number ∆Ca.

64

5.7. Conclusion

5.7. Conclusion

We have shown that the interplay of varying start up conditions and strongconfinement ratios can lead to qualitatively and quantitatively differentdroplet breakup conditions in stationary shear flows, unlike the stable equi-libria found for varying start up conditions [69] or the ones found for varyingdegrees of confinement [28]. Having investigated the effects of inertia, con-finement and flow topology, we conclude that the prtocol mismatch betweenthe shock and the relaxation method are due to a high degree of confine-ment for a droplet in a shear flow (α = 0.75). However, the breakup solutionfound via the relaxation method is only metastable, since it becomes un-stable in the case of a time-dependent, oscillatory shear flow. The protocolmismatch is thus solely due to an extra metastable solution in a stronglyconfined shear flow and it disappears in the presence of small perturbations,e.g. amplitude variations in an oscillatory shear flow, in accordance withthe uniqueness of the Stokes solution [28, 69]. We have also shown the de-pendency of the critical capillary number Cacr on the normalised oscillationfrequency ωf td in both oscillatory shear and elongational flows. In the caseof the elongational flow, Cacr increases with increasing ωf td, whereas nosimple functional dependence can be found for the oscillatory shear flow,since Cacr also depends on the flow start up and degree of confinement. Itwould be interesting to see whether the metastable solution can be foundin an experimental set up or whether it is too prone to perturbations tomanifests itself.

65

6. LBM scheme for asub-Kolmogorov droplet in HIT

In this chapter 1 we elaborate on the ghost node boundary flow introducedin chapter 3 and show that it can be used to model droplet dynamics onthe sub-Kolmogorov scale. Instead of using an analytical flow field, as wasthe case in chapters 4 and 5, we use a fully developed turbulent flow field,which we retrieve from pseudo-spectral simulation data of trajectories ofpassive tracers in homogeneous isotropic turbulence. We will outline theconditions for the local Reynolds number and the velocity profile on thesub-Kolmogorov scale and will test the algorithm in the case of a singlephase turbulent flow.

6.1. Introduction

Exact flow boundary conditions for LBM are a very useful tool for multiscalephysics simulations, as it is enabling scale separation in its very own nature.For example, we may model the dynamics of a droplet in a turbulent flowfield on the sub-Kolmogorov scale, without having to simulate the scale of thelargest turbulent eddies of size l0 (see chapter 2). Exact flow boundaries forLBM were first developed by [100] for 2D lattices and then extended to 3Dflows by [101,122,123] for perpendicular inlet and outlet flows. In [102] it isshown that exact boundary flow conditions can be implemented in 3D LBMsimulations for the D3Q19 lattice. The method presented in [102] is useful forno-slip LBM boundary flow conditions [124–126], as it is able to eliminatenumerical artefacts. Moreover, the D3Q19 exact boundary flow methodin [102] seems to be most adapt for LBM hybrid simulations [127–129], forexample coupling the results of a DNS solver for the incompressible NSEwith a multicomponent LBM simulation. Even though the scheme presented

1In preparation as: Milan, F., Sbragaglia, M., Biferale, L., Toschi, F., Sub-Kolmogorovdroplet dynamics in isotropic turbulence using a multiscale Lattice Boltzmann scheme

67

6. LBM scheme for a sub-Kolmogorov droplet in HIT

in [102] seems rather suitable in modelling a sub-Kolmogorov droplet inhomogeneous isotropic turbulence, we use the ghost node boundary flowmethod outlined in section 3.5. The main reason for this is that the methodin [102] cannot determine both the density ρb(x, t) and the velocity fieldv(x, t) at the lattice domain boundary. Since we require a constant densityratio of ρd/ρs for our hybrid simulations, we use the ghost node boundaryscheme, which does not set constraints to either the density nor the velocityfield at the boundary.

6.2. Structure of the algorithm

We have introduced the ghost node boundary flow method in section 3.5as an exact flow boundary method: the boundary scheme enables us toenforce a density field ρ(x, t) and u(x, t) at the boundaries at any LBMsimulation time step t. A concrete set up for the sub-Kolmogorov dropletin homogeneous isotropic turbulence is given in figure 6.2. A droplet withradius R in the undeformed stage is at the centre of a simulation box oflength Ls. The blue dots at the faces of the simulation box denote theghost nodes, which according to equation (3.24) contain the macroscopicboundary values, the density ρb(x, t) and the velocity u(x, t). As in previouschapters we deal with a density ratio of 1 and thus choose ρb(x, t) accordingly,whereby ρb = constant. However, the velocity field u(x, t) is now updatedvia pseudo-spectral (PS) data of passive trajectories in turbulent flows 2, seefigure 6.1. All the turbulent trajectories have a Taylor Reynolds number ofReλ = 420. We consider droplets below the Kolmogorov scale, i.e. R ηK ,as we can limit ourselves to a linear velocity profile in this case [14], wherewe use the notation Gij(t) ≡ ∂jvi(x, t) for the velocity gradient tensor. Thetypical velocity increments [6] of separation l of size |l| = l are

δvi(x, l) = vi(x + l)− vi(x) (6.1)

which can be approximated by

δvi(x, l) = vi(x)− vi(x)︸︷︷︸=0

+Gijlj +O(l2) (6.2)

2https://data.4tu.nl/repository/uuid:a64319d5-1735-4bf1-944b-8e9187e4b9d6

68

https://data.4tu.nl/repository/uuid:a64319d5-1735-4bf1-944b-8e9187e4b9d6

6.3. Validation results

-80

-60

-40

-20

0

20

40

60

80

0 1000 2000 3000 4000

Ve

locity g

rad

ien

t: G

xx

PS time steps

DNS: turbulent signal

Figure 6.1.: The DNS turbulent signal for the component Gxx of the velocity gradienttensor Gij . The data is taken from passive tracer trajectories in a homogeneous isotropicturbulent flow obtained via PS simulations.

for l ∼ R. Since the droplet’s centre of mass remains fixed, i.e. we arein the frame of reference of the droplet, we obtain the following velocityrelation in the case of a single phase flow:

v(0)

i (x, t) = Gij(t)xj (6.3)

which is a linear velocity profile in the distance x from the droplet’s centreof mass. With v(0)

i (x, t) we denote the turbulent velocity profile imposed bythe boundary, whereas vi(x, t) is the LBM obtained velocity profile givenby equation (3.8). Now we test whether our LBM scheme can relax to theimposed velocity field given in equation (6.3).


Now that we have established the approximations we make to the velocityprofile on the sub-Kolmogorv scale, see equation (6.3), we would like to createa pseudo-spectral LBM hybrid method, which takes the turbulent velocitygradients Gij(t) as an input. In order for the LBM model to yield accuratepredictions, we need to simulate a fully developed turbulent flow at the

69


s

Figure 6.2.: LBM simulation set up: the droplet of scale R in the simulation domainwith dimensions Ls×Ls×Ls. The turbulent flow is initialised at the boundary nodes andvisualised via streamlines coloured according to the velocity magnitude ‖v‖. The bluedots indicate the ghost nodes, which are crucial in coupling the pseudo-spectral turbulentflow data to the LBM simulation.

70


domain boundaries, which then evolves into the centre of the domain. Thisis achieved by using both the ghost node boundary flow scheme describedin section 3.5, which stores suitable equilibrium distribution functions onghost nodes, and the pseudo-spectral values for Gij(t). The simulation setup is shown in figure 6.2 with the length of the simulation box Ls and thediameter of the undeformed droplet 2R highlighted.

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0 50000 100000 150000 200000

Gxx (

lbu

)

Time steps (lbu)

DNS: rescaled

LBE

Figure 6.3.: Comparison between the velocity gradient of the pseudo-spectral code andthe LBM code after rescaling. Since, both curves, the DNS turbulent input data andthe LBM reproduced Gxx, overlap, LBM is able to qualitatively reproduce the turbulentvelocity gradient.

with the time evolution of the flow completely determined via the tur-bulent velocity gradient tensor Gij(t). Now we test the convergence of theLBM algorithm for the single phase, where we need to recover the linearprofile of the turbulent flow field, see equation (6.3), to a given accuracy.Thus, recovering the velocity field in equation (6.3) effectively means re-covering the turbulent velocity gradient Gij(t) of the pseudo-spectral DNSsimulations, as the velocity profile is linear. We choose an LBM samplingtime ∆t measured in lbu, which allows the LBM algorithm to relax to thevelocity value required by equation (6.3). In figure 6.3, we see that our LBMscheme reproduces indeed the turbulent signal Gij(t) we provided it with,since the original rescaled DNS signal for the Gxx(t) component converges,with the value for Gxx(t) obtained with LBM. The rescaling for Gij(t) is

71


Gij(t) 7→uLBM

Ls‖Gij(t)‖∞Gij(t) (6.4)

where ‖. . .‖∞ denotes the maximum norm and uLBM is a typical LBMvelocity scale for which the Mach number Ma ≤ 0.1. In order to choose theparameter ∆t we need to check the global convergence of the LBM velocityfield v(x, t) with the one imposed at the boundary, see equation (6.3). Thecorresponding L2-error is provided in figure 6.4 and is defined via:

0

0.05

0.1

0.15

0.2

100 200 300 400 500 600 700 800 900 1000

E(0

) (t)

t / ∆ t

Ls = 48, ∆ t = 100 (lbu)

Ls = 48, ∆ t = 200 (lbu)

Ls = 48, ∆ t = 500 (lbu)

Ls = 48, ∆ t = 1000 (lbu)

Figure 6.4.: Convergence of the time dependent LBM boundary scheme for a singlephase homogeneous and isotropic turbulent flow (domain size: 48× 48× 48). The timeaveraged global error can be estimated to be about 〈E(0)(t)〉t ∼ 0.05 for a chosen rangeof ∆t in terms of LBM time steps t (in lbu).

E(0)(t) =1

u(0)rms(t)

[1

L3s

∫ Ls

0dx

∫ Ls

0dy

∫ Ls

0dz (v(x, y, z, t)− v(0)(x, y, z, t))2

] 12

(6.5)

where v(x, y, z, t) and v(0)(x, y, z, t)) are the velocity fields of the LBMsimulation and the imposed turbulent linear velocity profile respectively.u(0)

rms(t) is the rms value of v(0)(x, y, z, t). Since we would like to choosea minimal ∆t, as to optimise run time, we choose ∆t = 200 (lbu), sincethis value yields a time averaged global error of 〈E(0)(t)〉t < 0.1. This is a

72

6.4. Conclusion

reasonable error threshold, since LBM simulations for droplet dynamics inoscillatory shear flows in chapter 4 yield a similar maximal error [74].

6.4. Conclusion

We have shown that similar to previous LBM flow boundary conditions [101,102] the LBM ghost node boundary flow method, see section 3.5, in conjunc-tion with pseudo-spectral turbulent flow data is able to reproduce temporalturbulent velocity profiles. The algorithm is only suited for droplets on thesub-Kolmogorov scale, where the linear profile of equation (6.3) is valid. Theturbulent gradient data is rescaled in order to adapt to varying domain sizesand to respect the Mach number limit of LBM simulations of incompressibleflows, i.e. Ma ≤ 0.1. Consequently, droplet dynamics and breakup can bemodelled fully resolved on the sub-Kolmogorov scale, which we will explorein chapter 7.

73

7. Dynamics and breakup of asub-Kolmogorov droplet in HIT

This chapter 1 deals with the dynamics and breakup of the sub-Kolmogorovdroplet in homogeneous and isotropic turbulence. We employ the ghostnode boundary flow method of chapter 3 and use pseudo-spectral data ofturbulent passive tracer trajectories to simulate a fully developed turbulentflow field, as outlined in chapter 6. Now we deal with a multicomponent flowonce again, where we place ourselves in the frame of reference of the droplet,i.e. the droplet remains fixed at the centre of the simulation domain and thesolvent flow is simulated around it. We investigate breakup predictions forseveral pseudo-spectral turbulent flow fields via the MM-model and comparethe results to the DNS-LBM hybrid method.

7.1. Introduction

The dynamics and breakup behaviour of immiscible droplets in laminar flowshave been studied extensively in the literature [25,26,29,37,46,103,105]. Lessis known about the dynamics of droplets in homogeneous isotropic turbulentflows, which pose the challenge of solving a multi-physics problem, sincethe flow properties of the turbulent scales have to be accurately transferredto the scale of the droplet [12,13]. Furthermore, the dynamics and breakupstatistics of sub-Kolmogorov droplets have also been studied extensively [15,17,18], which is of particular interest, since the viscous stresses dominate overthe inertial stresses, if the droplet size is smaller than the Kolmogorov scaleof the outer flow [14]. Both [17] and [18] make use of the MM-model [16] todescribe droplet dynamics in homogeneous and isotropic turbulence, whichonly accounts for ellipsoidal deformations and thus can model breakup onlyvia a cut off criteria. Our studies differ in such a way that we not only use the

1In preparation as: Milan, F., Sbragaglia, M., Biferale, L., Toschi, F., Sub-Kolmogorovdroplet dynamics in isotropic turbulence using a multiscale Lattice Boltzmann scheme

75

7. Dynamics and breakup of a Sub-Kolmogorov droplet in HIT

MM-model to measure the deformation of the droplet, but compare it withfully resolved LBM simulations using the SCMC diffuse interface method [77,78]. LBM in conjunction with SCMC has been extensively used in thefield of microfluidics and droplet breakup [12, 55, 83–90] making it a wellsuited tool to predict breakup behaviour with high accuracy. Our numericalmethod can be seen as a new hybrid approach to the sub-Kolmogorovdroplet dynamics: The simulation domain is of the order of the size of thedroplet and the surrounding turbulent flow is used as an input via an openboundary condition method [74], similar to [101, 102]. This enables us tostudy the deformation and breakup behaviour of a sub-Kolmogorov dropletin fully developed isotropic turbulence, which we can compare to the resultsobtained via the widely used MM-model.

7.2. Flow behaviour of the sub-Kolmogorov droplet

Chapter 6 described the outline of a hybrid DNS-LBM algorithm to simulatea turbulent solvent flow for droplets on the sub-Kolmogorov scale. Moreover,we have explained in section 2.4, that the local Reynolds number Re < 1 onthe sub-Kolmogorov scale, even though the Taylor Reynolds number Reλmay be several magnitudes larger, see equation (2.34). Consequently, weare very close to the Stokes limit of Re→ 0 and may neglect inertial effects.Thus, we can make use of a relation of Stokesian suspension dynamics [130]:the velocity field v(0)(x, t) of the undisturbed flow is combined with thevelocity field of the suspended particle vd(x, t), in our case a deformabledroplet, to yield the actual velocity of the flow 2:

va(x, t) = v(0)(x, t) + vd(x, t) (7.1)

In case of a weakly deformed, almost spherical droplet, the only possiblemotion of the suspended particle would be rotation, as translation is ruledout in our simulations, see chapter 6. A spherical droplet rotating withangular velocity ωd(t) would then modify the flow field by [130]

vd(x, t) = ωd(t)× x

(R

|x|

)3

(7.2)

2The equations are modified to include an explicit time dependency for the flow.

76

7.3. Breakup predictions: LBM vs MM

where |x| =√x · x, with x being the distance vector from the droplet’s

centre of mass. The actual velocity field is then given by

va(x, t) = v(0)(x, t) + ωd(t)× x

(R

|x|

)3

(7.3)

Although equation (7.3) seems like it is of great use to the study of sub-Kolmogorov droplets in homogeneous and isotropic turbulence [131], itshould be stressed, that it is only valid for weakly deformed droplets, whosemotion is purely restricted to rotation. Therefore, we do not use equa-tion (7.3) in our droplet simulations in the next chapter, where we deal withlarge amplitude deformations and breakup.

7.3. Breakup predictions: LBM vs MM

Since we have established the validity of the hybrid PS-LBM algorithm inchapter 6, we can investigate droplet dynamics and breakup for a singledroplet on the sub-Kolmogorov scale. Analogously to equation (2.8) we candefine a measure for the deformation DLBM for our LBM simulations:

DLBM(t) =

√1− Ω0(t)

Ω(t)(7.4)

where Ω(t) is the time-dependent surface area of the deformed dropletand Ω0(t) is the surface area of an undeformed spherical droplet of the samevolume as the deformed droplet. DLBM(t) represents a bounded measurefor droplet deformation, because DLBM(t) = 0, in case of minimal deforma-tion, for which Ω(t) = Ω0(t), and DLBM(t) → 1 for Ω(t) → ∞ in case ofa hypothetically arbitrary large surface area Ω(t). It should be noted thatDLBM(t) scales with the droplet length L(t), see chapter Chapter 5, for elon-gated droplets and thus DLBM(t) ∼ D, where D is the droplet deformationaccording to major and minor ellipsoidal axes, see equation (2.8). We startour simulations by small amplitude deformations, so that the droplet is onlyellipsoidally deformed, for which we expect the LBM and MM-model resultsto match, see figure 7.2. As we can see from figure 7.1, this is indeed the case,because both predictions for the deformation D, LBM and the MM-model,coincide for a maximum capillary number Camax Cacr. Figure 7.1 showsthe deformation curves for five separate turbulent flows predicted by the

77


0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

D

Camax

LBM: flow 1MM: flow 1MM: flow 2MM: flow 3MM: flow 4MM: flow 5Taylor

Cacr

Figure 7.1.: MM-model predictions for the ellipsoidal droplet deformation D as a functionof Camax for various turbulent flows. The critical capillary number Cacr is estimated viaa cut off for D in the MM-model. The Taylor deformation law, see equation (2.9), isshown for comparison in the low Camax limit. We can see that the LBM results coincidewith the one of the MM-model, up until a maximum capillary number Camax ≈ 0.291,where the droplet is no longer ellipsoidally deformed, which causes the deviations fromthe MM-model predictions, see figures 7.3 and 7.4. The critical capillary number obtainedvia LBM is Cacr = 0.286± 0.006, which is lower than the one predicted in the MM-model,which is lies in the range CaMM

cr = 0.58±0.05. Therefore, it appears that the MM-model isnot able to predict the correct critical capillary number Cacr for sub-Kolmogorov dropletbreakup.

78

7.4. Conclusion

MM-model and one produced via the hybrid DNS-LBM simulations. Wehave chosen five different turbulent flows for the MM-model predictionsto account for the irregularity of turbulence. The MM-model deformationcurves and the LBM curve collapse for the intermediate range of Camax untilbreakup occurs for the LBM curve at around Camax ≈ 0.291, see figures 7.3and 7.4. Figure 7.3 shows the elongated droplet for a maximal capillarynumber Camax ≈ Cacr, just before and after breakup. Interestingly, thedroplet is now elongated in a similar way to the one in figure 5.1. Figure 7.4shows the same droplet breakup as figure 7.4 with the velocity field shownvia streamlines. We can see that the turbulent flow possesses a significantrotational part in the instance of the breakup. Let us consider the deforma-tion diagram of figure 7.1 again: LBM and the MM-model predict two verydifferent ranges for the critical capillary number Cacr. Firstly, this is due tothe fact, that breakup in the MM-model is determined via a cut off in thedeformation D, as the model does not account for a breakup mechanism perse. Secondly, the MM-model can only model ellipsoidally deformed droplets,even in the case of large amplitude flows. Therefore, the MM-model is un-able to model the elongated droplet of the DNS-LBM hybrid simulations,shown in figure 7.3, and thus cannot give a very accurate prediction for thecritical capillary number Cacr. The MM-model prediction for the criticalcapillary number is CaMM

cr = 0.58±0.05, where the error has been estimatedvia the spread of the MM deformation curves in figure 7.1. LBM predictsa lower range of Cacr = 0.286 ± 0.006, which is more accurate due to thefully resolved LBM SCMC model, even though the error might be larger,if more turbulent flow trajectories were considered, as has been the casefor the MM-model in figure 7.1. The LBM error in Cacr is estimated viaa step in the maximal critical capillary number ∆Camax ≈ 0.011. ∆Camax

represents the change in capillary number between individual data pointsin figure 7.1.

7.4. Conclusion

The DNS-LBM hybrid scheme for sub-Kolmogorov droplets in fully devel-oped homogeneous and isotropic turbulence has been used to model dropletdeformation and breakup. The LBM results were also compared to MM-model predictions of droplet deformation and breakup. We have found thatLBM predicts a significantly lower critical capillary number Cacr than the

79


MM-model, see figure 7.1. The reason for this is probably two-fold: firstlybreakup is chosen via a cut off procedure for the deformation parameterD inthe MM-model and secondly the droplet is far from an ellipsoidal shape in theregion of maximum capillary numbers Camax ≤ Cacr, see figures 7.3 and 7.4.This suggests, that even though the MM-model and DNS-LBM predictionsagree very well in the capillary number region Camax Cacr, accuratepredictions of sub-Kolmogorov droplet breakup can only be carried out viafully resolved simulations, such as the DNS-LBM hybrid algorithm, insteadof phenomenological models such as the MM-model [17, 18]. It would beinteresting to investigate droplet deformation statistics of Sub-Kolmogorovdroplets with the DNS-LBM hybrid algorithm and once again use the MM-model for comparison [17, 18]. In addition, it would be interesting to seethe range of critical capillary numbers Cacr predicted by the DNS-LBMhybrid model by extending the LBM results of droplet breakup in figure 7.1for various turbulent flow fields. Finally, a further extension would be theaddition of the velocity field modification outlined in equation (7.3) forweakly deformed rotating sub-Kolmogorov droplets.

80

7.4. Conclusion

Figure 7.2.: Sub-Kolmogorov droplet for a capillary number Camax Cacr with andwithout the velocity field shown via streamlines. The droplet is ellipsoidally deformed,which is the regime where LBM and MM predictions coincide.

81


Figure 7.3.: Sub-Kolmogorov droplet for a capillary number Camax ≈ Cacr before andafter breakup. The elongated droplet shape indicates that substantial deviations from theMM-model predictions might be possible.

82

7.4. Conclusion

Figure 7.4.: Sub-Kolmogorov droplet with the velocity field indicated via streamlinesfor a capillary number Camax ≈ Cacr before and after breakup. We see that the elongatedshape of the droplet is due to both a strong strain and rotational part in the turbulentflow field at this time instance.

83

8. Conclusion and outlook

The main two goals of this thesis were droplet dynamics and breakup inconfined time-dependent laminar flows and the development of a DNS-LBMhybrid algorithm to study sub-Kolmogorov droplet deformation and breakupin homogeneous isotropic turbulent flows. We will summarise and explainthe results of those two goals in the following sections.

8.1. Droplets in laminar time-dependent flows

As was mentioned in the introduction, first attempts to model droplet de-formation, both in unconfined [19–21] and confined systems [39, 40], arebased on the Stokes flow equations, see equation (2.4). Thus, we tested theghost boundary flow algorithm of section 3.5 for flows close to the Stokeslimit, with a simple time-dependence added to the flow in the form of anoscillatory shear rate, see chapter 4. We have tested the validity of the ghostboundary flow algorithm for a simple single phase time-dependent flowproblem, a flow created by an oscillating moving wall in an infinitely longchannel [91]. After reproducing the exact solution given in [91] with LBM,we analysed the deformation of a single droplet in the channel flow, withtwo oscillating walls, now producing a time-dependent shear rate. The LBMresults were tested with predictions of the MM-model [16,41,42] and thatof a linearised MM-model [73]. Moreover, we investigated the “transparencyeffect”, which occurs for a specific choice of the oscillation shear frequencyin relation to the droplet’s relaxation time. In chapter 5 we investigatedlarge scale droplet deformation and breakup of a confined droplet in anoscillatory shear flow. A striking question which arose, was the reason for amismatch between two different flow start up protocols, the “shock method”and the “relaxation method”. This led us to investigate the dependencyof this mismatch on inertia [46, 69], degree of confinement [28] and flowtopology [67,68]. We concluded, that the reason for the mismatch was dueto a high degree of confinement, α ≥ 0.75 and would disappear under the

85

8. Conclusions and outlook

influence of small scale flow perturbations, such as the time-dependency ofan oscillatory shear flow. After having investigated rather interesting casesof time-dependency effects in laminar multicomponent flows, we moved tothe turbulent regime to analyse the dynamics of sub-Kolmogorov droplets.

8.2. Sub-Kolmogorov droplets in turbulent flows

Before starting our investigation of sub-Kolmogorov droplets in homoge-neous isotropic turbulent flows, we outlined the structure of the ghost nodeboundary flow algorithm of section 3.5 for the use in HIT on the sub-Kolmogorov scale in chapter 6 and compared it briefly to other exact bound-ary flow algorithms [100–102] and hybrid methods [127–129]. We explainedthe way the pseudo-spectral data of passive tracer trajectories in HIT wasused to mimic an exact flow at the LBM simulation domain boundary andshowed that we can reproduce the turbulent flow in single phase LBM sim-ulations with the newly developed DNS-LBM hybrid scheme. Eventually,chapter 7 saw the DNS-LBM hybrid method of chapter 6 be put to use.Firstly, we mentioned that sub-Kolmogorov droplet dynamics are challeng-ing to analyse in a variety of settings [14, 15] and that the MM-model isthe method of choice to model droplets in HIT [17,18]. However, we havedemonstrated that the fully resolved DNS-LBM hybrid method and the MM-model yield different results for large amplitude sub-Kolmogorov dropletdeformation and break up. Therefore, we may conclude that, one shouldprefer the DNS-LBM hybrid simulations over the MM-model, if accuratepredictions about breakup on the sub-Kolmogorov scale are essential.

8.3. Outlook

This work on droplet dynamics in time-dependent and turbulent flows hasled to many interesting questions to be answered along the way. Firstly, theprotocol mismatch in chapter 5 has never been observed in the literaturebefore. It would be interesting to see, if an experimental setup with a suffi-ciently isolated and stable shear flow and smooth flow startup conditionswas able to reproduce the protocol mismatch. Furthermore, a detailed com-parison of sub-Kolmogorov droplet dynamics statistics using the DNS-LBMhybrid scheme vs the MM-model would be of interest as well, in additionwith further DNS-LBM breakup statistics for the results shown in figure 7.1.

86

8.3. Outlook

A further extension of the work could be to modify the velocity field inthe DNS-LBM hybrid setup according to equation 7.3 for weakly deformedsub-Kolmogorov droplets.

87

A. Numerical solution of theMM-model

The droplet deformation equation of the MM-model [16], given in equa-tion (2.13), is a non-linear equation in the droplet deformation tensor Mij

and was thus solved numerically. We will briefly outline the numerical solverused for the MM-model in this chapter and discuss its validation tests fordifferent solvent flows, for which analytical solutions of the MM-model of [16]are used. Before we start to discretising the MM-model equation (2.13), weperform a non-dimensionalisation

dMij

dt′=[f2(S′ikMkj +MikS

′kj) + Ω′ikMkj −MikΩ

′kj

]− f1

Ca

(Mij − 3

IIIMIIM

δij

)(A.1)

where we have rescaled the shear tensor with the shear rate G, so thatS′ij = Sij/G, Ω′ij = Ωij/G and t′ = t/G. The constants f1 and f2 of therescaled MM equation (A.1) are given by

f1(χ) =40(χ+ 1)

(2χ+ 3)(19χ+ 16)(A.2)

f2(χ) =5

2χ+ 3(A.3)

where f1 and f2 are soley dependent on the viscous ratio χ 1. The rescaledMaffetone-Minale equation (A.1) is tested for three specific shear flowsin [16]. We would like to validate those cases numerically with a Runge-Kutta 4 (RK4) integration scheme. In order to solve equation (A.1) with the

1The corresponding values for MM-unbounded in equation (2.16) differ from the valuesgiven in equation (A.2) via corrective terms in the capillary number Ca. Even thoughthe parameters of equation (2.16) increase the accuracy of the model, the analyticalsolutions of the MM-model in this chapter use the non-corrective parameters given inequation (A.2)

89

A. Numerical solution of the MM-model

parameters given in equation (A.2), we consider an RK4 scheme for a firstorder dynamical differential equation x = f(x, t), where the n-th timestepis given by [132]

k1 = f(xn, tn) (A.4)

k2 = f(tn +

∆t

2, xn +

k1

2

)(A.5)

k3 = f(tn +

∆t

2, xn +

k2

2

)(A.6)

k4 = f(tn + ∆t, xn + k3

)(A.7)

xn+1 = xn +∆t

6

(k1 + 2k2 + 2k3 + k4

)+O(∆t5) (A.8)

with ∆t being the discrete time step for the RK4 numerical integrator.With the help of this solver we can obtain a numerical value of the dropletdeformation parameter D via the time evolution of the morphology tensorgiven in equation (A.1). We will apply three different types of solvent flowsto the MM-model equation (A.1) in the following sections and compare thesolution of the RK4 numerical solver to the analytical solutions providedby [16].

A.1. Simple shear flow

At first let us consider a simple shear flow with strain rate

S′ij =1

2

0 1 01 0 00 0 0

(A.9)

and rotation

Ω′ij =1

2

0 1 0−1 0 00 0 0

(A.10)

According to [16] the steady-state values for the semi-axes of the ellipsoidare given by

90

A.2. Uniaxial extensional flow

L2 =f2

1 + Ca2 + f2Ca√f2

1 + Ca2

(f21 + Ca2)

13 (f2

1 + Ca2 − f22Ca

2)23

(A.11)

B2 =f2

1 + Ca2 − f2Ca√f2

1 + Ca2

(f21 + Ca2)

13 (f2

1 + Ca2 − f22Ca

2)23

(A.12)

W 2 =f2

1 + Ca2 − f22Ca

2

(f21 + Ca2)

13 (f2

1 + Ca2 − f22Ca

2)23

(A.13)

yielding a deformation parameter of

Dshear =

√f2

1 + Ca2 −√f2

1 + Ca2 − f22Ca

2

f2Ca(A.14)

A.2. Uniaxial extensional flow

Secondly we check the model against a uniaxial extensional flow [16] withstrain

S′ij =1

2

2 0 00 −1 00 0 −1

(A.15)

and an identically vanishing rotation

Ω′ij = 0 (A.16)

with semi-axes

L2 =

(f1 + f2Ca

f1 − 2f2Ca

) 23

(A.17)

B2 = W 2 =

(f1 − 2f2Ca

f1 + f2Ca

) 13

(A.18)

giving a deformation parameter

Duniaxial =2f1 − f2Ca− 2

√(f1 + f2Ca) (f1 − 2f2Ca)

3f2Ca(A.19)

91

A. Numerical solution of the MM-model

A.3. Planar hyperbolic flow

At last we test the MM-model for a planar hyperbolic flow

S′ij =

1 0 00 −1 00 0 0

(A.20)

and an identically vanishing rotation

Ω′ij = 0 (A.21)

with semi-axes

L2 =(f2

1 + 2f1f2Ca)13

(f1 − 2f2Ca)23

(A.22)

B2 =(f2

1 − 2f1f2Ca)13

(f1 + 2f2Ca)23

(A.23)

W 2 =(f2

1 − 4f22Ca

2)13

f23

1

(A.24)

and a deformation parameter

Dplanar =f1 −

√f2

1 − 4f22Ca

2

2f2Ca(A.25)

The corresponding numerical validation tests are given in figure A.1, fromwhich we can see that the numerical solver is able to accurately reproducethe analytical solutions to the three different types of flow fields for variousviscosity ratios χ.

92

A.3. Planar hyperbolic flow

0

0.05

0.1

0.15

0.2

0.25

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D

Ca [Simple shear χ = 0.1]

Numerical solutionAnalytical solution

0

0.05

0.1

0.15

0.2

0.25

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D



0

0.05

0.1

0.15

0.2

0.25

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D

Ca [Uniaxial extensional χ = 0.1]


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D

Ca [Planar hyperbolic χ = 0.1]


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

D



Figure A.1.: Numerical validation of the numerical RK4 solver for the MM deformationequation (A.1), for simple shear, uniaxial extensional and planar hyperbolic flows.

93

B. A linearised MM-model solutionfor an oscillatory shear flow

In chapter 4 we investigated the deformation of a single confined dropletin an oscillatory shear flow. We performed LBM simulations and comparedtheir results to both a numerical and a perturbative solution of the MM-model. This chapter gives a brief outline of the ideas behind the linearisedperturbative MM-model used during the discussion in chapter 4. The mor-phology tensor Mij in the MM-model equation (4.11) may be expressed ina perturbation series in the time-dependent capillary number Ca(t) [73]:

Mij(t) = δij + Ca(t)M(1)ij (t) +O(Ca2(t)) (B.1)

where M(1)ij (t) denotes a first order correction to the morphology tensor

of an undeformed droplet δij and the time t is given in units of the dropletrelaxation time td. The time evolution of the capillary number is determinedvia the oscillatory shear rate, which we can express in complex form as

Ca(t) = iCamaxeiωtdt (B.2)

Using the relations in equations (B.1) and (B.2) and the initial condition

M(1)ij (0) = 0, we obtain an evolution equation for the first order off-diagonal

elements of M(1)ij (t):

dM(1)xz (t)

dt= f2 − (f1 + iωtd)M

(1)xz (t) +O(Ca2(t)) (B.3)

where M(1)xz (t) ∼ Ca(t) is the off-diagonal element of Mij in accordance

with the applied shear in the x-z-plane, see figure 4.3. The solution toequation (B.3) is then given by

M (1)xz (t) =

f2

(1− e−f1t−iωtdt

)f1 + iωtd

+O(Ca2(t)) (B.4)

95

B. A linearised MM-model solution for an oscillatory shear flow

from which we can diagonalise the linearised morphology tensor Mij(t) toobtain up to first order in the maximal capillary number Camax the complexeigenvalues L2

c , W2c and B2

c of Mij(t):

L2c = 1 + Camaxf2

(eiωtdt − e−f1t

ωtd − if1

)+O(Ca2

max)

W 2c = 1− Camaxf2

(eiωtdt − e−f1t

ωtd − if1

)+O(Ca2

max)

B2c = 1 +O(Ca2

max) (B.5)

The transient part e−f1t can be ignored for sufficiently large t. Aftertaking the real part of the complex eigenvalues of Mij we obtain the valuesgiven in equation (4.12).

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109

Summary

This thesis deals with the numerical study of the dynamics and instabilityof a single droplet in complex flows, such as laminar time-dependent andturbulent flows. The numerical model to be used is the Lattice BoltzmannMethod (LBM), which is able to accurately simulate fully resolved multicom-ponent flows with the help of a diffuse interface method. We have developeda novel LBM boundary scheme, which enables us to study droplets in tur-bulent flows on the sub-Kolmogorov scale. Turbulent droplet dynamics isa multiscale problem, because information from both the droplet and thelargest turbulent scales are needed. Therefore the LBM boundary schememust simulate arbitrary velocity fields at the domain boundary, which incase of this thesis are velocity fields of laminar oscillatory flows and ho-mogenous and isotropic turbulent flows. Moreover, the LBM results arecompared with the solutions to a phenomenological droplet deformationmodel, the Maffettone-Minale model (MM-model). In order to test the newLBM boundary scheme, we considered the case of droplet dynamics in aconfined and oscillatory shear flow. We find that the LBM results and theresults of a confined version of the MM-model match well. Furthermore,we considered the case of high frequency oscillatory shear flows in relationto the droplet relaxation time and analysed the “transparency effect”, forwhich the droplet is only marginally deformed, even in the case of largeflow amplitudes. Additionally, we investigated LBM breakup predictionsof a droplet in an oscillatory confined flow, in relation to the oscillatoryfrequency, the degree of confinement, inertial effects, flow start up conditionsand flow topology. The LBM breakup simulations suggest that breakup inconfined stationary shear flows may lead to different breakup conditionsfor different flow start up conditions in the presence of strong confinement.This mismatch between the LBM breakup results vanishes in the presenceof small perturbations, such as small flow amplitude variations in an oscilla-tory shear flow. The newly developed LBM boundary scheme would allowfor the introduction of an external turbulent flow field. This is achievedby using data of passive tracer trajectories in homogenous and isotropic

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Summary

turbulent flows obtained via pseudo-spectral simulations. After rescalingand finding a suitable sampling algorithm of the turbulent flow data, wecan recover the correct single phase linear turbulent velocity profile. Thismultiscale hybrid LBM algorithm is now suitable to predict sub-Kolmogorovdroplet dynamics and breakup. We find, that the results of both LBM andthe MM-model for the deformation dynamics agree well with each other forsmall capillary numbers. However, LBM yields different deformation andbreakup predictions than the one of the MM-model in the large capillarynumber regime. This is due to the fact that the LBM hybrid scheme is ableto model a fully resolved turbulent multicomponent flow, contrary to theMM-model simulations. This work investigates the effect of explicit flowtime-dependencies for droplet deformation in confined shear flows, with thehelp of new LBM boundary method. Furthermore, this boundary methodis extended to a novel multiscale hybrid LBM algorithm used to model afully resolved sub-Kolmogorov droplet in homogeneous and isotropic tur-bulence. The work in this thesis also demonstrates the versatility of LBMand DNS-LBM hybrid algorithms for both multiphase and multicomponentflows in particular.

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Acknowledgement

“The whole is more than the sum of its parts.” Even though this thesisdisplays the findings of my PhD research, it does not provide the completepicture of my development during my PhD research programme both pro-fessionally and personally. I would like to thank everyone who has been apart of it and made the four years of PhD research that much more enrich-ing. First of all I thank my promoters, Prof. Luca Biferale, Prof. MauroSbragaglia and Prof. Federico Toschi. Through your expert knowledge andguidance I was able to tackle challenging numerical and multiphysics prob-lems in my research project. In particular, I am grateful for the in-depthdiscussions on several problems regarding the research direction and contentof this project. And last but not least, thank you for encouraging me to goto several schools and trainings to broaden my knowledge and giving methe opportunity to present my research results at international conferencesworldwide. My gratitude also goes to Prof. Matthias Erhardt and Prof. An-dreas Bartel who gave me the possibility to present my work at a differentangle to a numerics audience and with whom I had stimulating discussionsabout Lattice Boltzmann boundary flows. I also thank the PhD defensecommittee members, Prof. Patrick Anderson, Prof. Luca Brandt and Dr.Wim-Paul Breugem for reviewing this thesis. I also thank the organisersof my European Joint Doctorate programme HPC-LEAP and the MarieSk lodowska-Curie Actions for making the research presented here possible.

I would particularly like to thank my fellow HPC-LEAP PhD researchers,Alessandro, Srijit, Slava, Simone, Wenping, Andy, Aurora, Salvatore, Ibo,Thomas, Teo and Vanessa. Special thanks goes to Guillaume, who wouldnever lose his sense of optimism in stressful times and always was a source ofencouragement to the other HPC-LEAP fellows. Starting a new job abroadis by no means easy, I was thus glad to be at the same institution as Georgios,a great friend with whom the start of the project was just a little easier.Colleagues don’t necessarily become close friends, but work just feels muchless of a burden, if they do. I am very happy that my research project was

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Acknowledgement

closely linked to the one of Xiao. Not only was I able to learn a lot fromyou, be it programming related or a first understanding of Chinese culture,but also developed a very close friendship.

My gratitude also goes to my colleagues and friends at Rome Tor Vergata,Anupam, Riccardo, Massimo, Matteo, Claudio, Roberto, Simona and Tadeja.I thank Michele especially, who was not tired to help out every new PhDstudent with the encumbering Roman bureaucracy and made sure that theLeapers were feeling comfortable in Rome, even if some of them just droppedin for a conference. In particular, I thank Nhut and Abhi for organising theevening football matches. I would also like to thank Olmo, Maria, Nesrine,Daniele and Henriet for organising great house parties and always havingan upbeat dynamics, no matter where or when.

A big thank you goes out to Marjan Rodenburg, who with great care anddedication handled every problem my colleagues and I were throwing at her.I would also like to thank my friends and colleagues at the department, Ab-hineet, Andres, Andrej, Alessandro, Arnab, Cosimo, Gianluca, Gianmarco,Giorgia, Hadi, Haijing, Han, Jonathan, Karun, Kim, Matteo, Rik, Marlies,Marteen, Sam, Sebastian, Steven, Vignesh and Twan. Special thanks goesto my officemates Pinaki and Abheeti, it was a privilege to share an officewith you and I will dearly miss the office banter and I’m especially thank-ful for giving me a glimpse into Indian culture during the Bangalore trip.Furthermore, my gratitude goes to Bijan and Lenin. It was just that mucheasier to settle into Eindhoven with you, be it because of the coffee breakdiscussions, campus walks or the weekend get-togethers.

I also thank my (mostly) non-academic friends in Eindhoven, Alejo, Calin,Marco, Mohit, Oscar, Santosh, Stefan, Ruxi and Ivan. I would also like tothank my family, my parents and my sisters, for their unwavering supportduring my PhD, especially during the more difficult times. And finally Iam thankful for your understanding and loving support Diana, particularlyduring the stressful times of the writing process.

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Curriculum Vitae

Felix Milan was born on November 22, 1990 in Bamberg, Germany. Hestudied physics at the University of Bayreuth with a focus on theoreticaland computational physics in his final research project on applications ofdensity functional theory to organic solar cells. After obtaining his B.Sc.degree, he moved to Leiden University for his M.Sc. studies in physics witha research focus in theoretical physics. He graduated cum laude with atheoretical and computational research project on chiral phase transitionsof frustrated nematic systems. In 2015 he obtained a Marie-Curie fellowshipto start in October 2015 a four year Joint European Doctorate in theoreticaland applied physics at the University of Rome Tor Vergata and EindhovenUniversity of Technology. The results of this research programme are pre-sented in this dissertation. He works at ASML, Veldhoven, as a functionalmetrology design engineer since October 2019.

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Cover Illustration

Snapshot of a deformed sub-Kolmogorov droplet in a homogenous andisotropic turbulent flow. The contour of the droplet is shown in light grey,whereas the flow is visualised via streamlines, which are colour coded interms of the velocity magnitude, increasing from blue to red. The simulationparameters (in lbu) are: radius of the initially undeformed droplet R = 40,the domain length Ls = 288 and Reλ = 420.

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List of publications and conferences

Journal articles

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Sub-Kolmogorovdroplet dynamics in isotropic turbulence using a multiscale LatticeBoltzmann scheme, (submitted to Journal of Computational Science)

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Lattice Boltzmannsimulations of droplet breakup in confined and time-dependent flows,(submitted to Physical Review Fluids)

• F. Milan, M. Sbragaglia, L. Biferale and F. Toschi, Lattice Boltz-mann simulations of droplet dynamics in time-dependent flows Eu-ropean Physical Journal E, 2018, 41:6. https://doi.org/10.1140/epje/i2018-11613-0

Conferences contributions

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multi-scale LBMsimulations of droplets in time-dependent flows, Burgers Symposium2017, Lunteren, the Netherlands, May 2017

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multi-scale LBMsimulations of droplets in time-dependent flows, 26th InternationalConference on Discrete Simulation of Fluid Dynamics, Erlangen, Ger-many, July 2017

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multi-scale LBMsimulations of droplets in time-dependent flows, 70th Annual Meetingof the American Physical Society Division of Fluid Dynamics, Denver,USA, November 2017

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https://doi.org/10.1140/epje/i2018-11613-0

https://doi.org/10.1140/epje/i2018-11613-0

List of publications and conferences

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Lattice Boltzmannsimulations of droplet dynamics in time-dependent flows (poster),Physics@Veldhoven 2018, Veldhoven, the Netherlands, January 2018

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multi-scale LBMsimulations of droplets in generic time-dependent flows, COST ActionMP1305: Flowing Matter 2018, Lisbon, Portugal, February 2018

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Lattice Boltzmannstudy of droplet dynamics and break-up in generic time-dependentflows, 27th International Conference on Discrete Simulation of FluidDynamics, Worcester MA, USA, June 2018

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Lattice Boltzmannstudy of droplet dynamics and break-up in generic time-dependentflows, HPC-LEAP Concluding Conference, Cambridge, UK, July 2018

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Lattice Boltzmannstudy of droplet dynamics and break-up in generic time-dependentflows, 71st Annual Meeting of the American Physical Society Divisionof Fluid Dynamics, Atlanta, USA, November 2018

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multi-scale LatticeBoltzmann simulations of droplets in time-dependent and turbulentflows (poster), Physics@Veldhoven 2019, Veldhoven, the Netherlands,January 2019

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multiscale LatticeBoltzmann simulations of droplet dynamics in turbulent flows, BurgersSymposium 2019, Lunteren, the Netherlands, May 2019

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multiscale LatticeBoltzmann simulations of droplet dynamics in turbulent flows, 28thInternational Conference on Discrete Simulation of Fluid Dynamics,Bangalore, India, July 2019

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Multiscale LatticeBoltzmann simulations of droplet dynamics in turbulent flows, 17thEuropean Turbulence Conference, Turin, Italy, September 2019

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Invited talks

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, Lattice Boltzmannsimulations of droplet dynamics in time-dependent flows, Universitadegli Studi Niccolo Cusano, Rome, Italy, December 2017

• F. Milan, L. Biferale, M. Sbragaglia and F. Toschi, The Lattice Boltz-mann Method and applications for time-dependent multi componentflows, Bergische Universitat Wuppertal, Wuppertal, Germany, March2018

Journal articles out of scope of this thesis

• C. Schwarz, F. Milan, T. Hahn, M. Reichenberger, S. Kummel andA. Kohler, Ground State Bleaching at Donor-Acceptor Interfaces, Ad-vanced Functional Materials, 2014, 24, 64396448, 10.1002/adfm.201400297

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